Chapter 5. Circles

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Circles are special cases of ellipses(we will study these the Chapter 7). For circles the length of major and minor axes of an ellipse are equal. Equivalently, a circle can be defined as locus of a point whose distance from a fixed center remains constant.

Consider a point $(h, k)$ as the center, the distance called radius as $a$ and the point $(x, y)$ then the general equation of the locus can be written as

Figure 5.1. A circle

(x - h)^{2} + (y - k)^{2} = a^{2}

(5.1)

If we take the origin as the center then this equation reduces to

x^{2} + y^{2} = a^{2}

(5.2)

Figure 5.2. A circle with Center at Origin

In the last chapter we found the condition for general equation of second degree to represent a pair of straight lines. Here, we will find the condition for it to represent a circle.

Recall that general equaiton of second degree in $x$ and $y$ is given by $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ and equation of circle is $(x - α)^{2} + (y - β)^{2} = r^{2}$ .

Comparing the coefficients, we find that, $\frac{1}{a} = \frac{1}{b} = \frac{0}{h} = - \frac{α}{g} = - \frac{β}{f} = \frac{α^{2} + β^{2} - r^{2}}{c^{2}}$ , hence, coeff. of $x^{2}$ should be equal to coefficient of $y^{2}$ and coeff. of $x y$ should be zero.

Thus, general equation in second degree which represents a circle is $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ $(x + g)^{2} + (y + f^{2}) = g^{2} + f^{2} - c$ (5.3)

which will have center at $(- g, - f)$ and radius equal to $\sqrt{g^{2} + f^{2} - c}$ .

5.1. Special Cases

Case I: The circle passes through the origin.

Figure 5.3. A circle through Origin

The equation of the circle is $(x - α)^{2} + (y - β)^{2} = a^{2}$ . Since it passes through the origin, therefore, $(0 - α)^{2} + (0 - β)^{2} = a^{2}$ $α^{2} + β^{2} = a^{2}$ (5.4)

If we consider general equation then we find that $c = 0$ , thus the equation of circle is given by $x^{2} + y^{2} + 2 g x + 2 f y = 0$ (5.5)

Case II: The circle touches the $x$ -axis.

Figure 5.4. A circle touching x-axis

We see that $C L = | β |$ so we can write that $(x - α)^{2} + (y - β)^{2} = β^{2}$ (5.6)

In case we write it in form of general equation then $g^{2} + f^{2} - c = f^{2}$ . Thus, the equation of circle becomes

x^{2} + y^{2} + 2 g x + 2 f y + g^{2} = 0

(5.7)

Case III: The circle touches the $y$ -axis.

We see that $C L = | α |$ so we can write that $(x - α)^{2} + (y - β)^{2} = α^{2}$ (5.8) In case we write it in form of general equation then $g^{2} + f^{2} - c = g^{2}$ . Thus, the equation of circle becomes $x^{2} + y^{2} + 2 g x + 2 f y + f^{2} = 0$ (5.9)

Figure 5.5. A circle touching y-axis

Case III: The circle touches both the axes.

We first consider the case of first quadrant.

Figure 5.6. A circle touching both the axes

In this case $α = β = a$ , and hence, the equation becomes $(x - a)^{2} + (y - a)^{2} = a^{2}$ (5.10)

Similarly, in 2nd, 3rd and 4th quadrant the equation would be $(x + a)^{2} + (y - a)^{2} = a^{2}$ , $(x + a)^{2} + (y + a)^{2} = a^{2}$ , and $(x - a)^{2} + (y + a)^{2} = a^{2}$ .

5.2. Circle on a Diameter

We will find equation of a circle one of whose diameters has endpoints as $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ .

Figure 5.7. A circle about a diameter

Let $A (x_{1}, y_{1})$ and $B (x_{2}, y_{2})$ be the endpoints of the diameter $A b$ of the circle as shown. Also, let $P (α, β)$ on the circle. We know from geometry that $A B$ will subtend a right-angle on $P$ .

Slope of $A P = \frac{β - y_{1}}{α - x_{1}}$ , and slope of $B P = \frac{β - y_{2}}{α - x_{2}}$

Since these two lines are perpendicular to each other, we have $\frac{β - y_{1}}{α - x_{1}} . \frac{β - y_{2}}{α - x_{2}} = - 1$

$\Rightarrow (α - x_{1}) (α - x_{2}) + (β - y_{1}) (β - y_{2}) = 0$

Thus, equation of circle would be $(x - x_{1}) (x - x_{2}) + (y - y_{1}) (y - y_{2}) = 0$ (5.11)

5.3. Parametric Form of a Circle

We have the equation of the circle as $(x - α)^{2} + (y - β)^{2} = a^{2}$ then any point on the circle in parametric form can be given by $(α + a \cos θ, β + a \sin θ)$ .

If the center of the circle is at the origin i.e. the equation is $x^{2} + y^{2} = a^{2}$ then the point's coordiantes changes to $(a \cos θ, b \sin θ)$ . The point is often referred as point “ $θ$ ”.

5.4. Position of a Point w.r.t. a Circle

Consider any point $(x_{1}, y_{1})$ . For any circle the point will be inside, outside or on the circle as $(x_{1} - α)^{2} + (y_{1} - β)^{2} > a^{2}, < a^{2}$ or $= a^{2}$ .

For general second degree equation the condition would be $(x_{1} + g)^{2} + (y_{1} + f)^{2} > g^{2} + f^{2} - c, < g^{2} + f^{2} - c$ or $= g^{2} + f^{2} - c$ .

5.5. Intersection of a Line and a Circle

We know that one of the equations of a straight line is $\frac{x - α}{\cos θ} = \frac{y - β}{\sin θ}$ . This equation represents a line passing through a point $P (α, β)$ making an angle of $θ$ with positive direction of $x$ -axis. Let this ratio be equal to $r$ , where $r$ is the algebraic distance of the point $(x, y)$ from $P (α, β)$ . So the coordiantes on this line are given by $(α + 𝑟 \cos θ, β + 𝑟 \sin θ)$ .

If this point lies on the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ , then $(α + r \cos θ)^{2} + (β + r \sin θ)^{2} + 2 g (α + r \cos θ) + 2 g (β + r \sin θ) + c = 0$ i.e. $r^{2} + 2 r (α \cos θ + β \sin θ + g \cos θ + f \sin θ) + α^{2} + β^{2} + 2 g α + 2 f β + c = 0$ (5.12)

This is a quadratic equaiton in $r$ , and hence, the line through $P$ meets the circle at two points $A$ and $N$ . Then we see that $P A . P B = α^{2} + β^{2} + 2 g α + 2 f β + c$ , which is independent of $θ$ , i.e. the direction of the line.

This we know from geometry that from a point $P$ a secant drawn to cut the circle in two points $A$ and $B$ , the product of the distances $P A$ and $P B$ is constant.

5.6. Tangents and Normals

Figure 5.8. Tanngents and normals

Consider two points $P$ and $Q$ on a curve. The position to which the line $P Q$ tends as $Q$ becomes closer to the point $P$ i.e. the limiting position of chord $P Q$ as $Q$ tends to $P$ along the curve is called the tangent to the curve at point $P$ . The point $P$ is called the point of contact of the tangent.

We will find equation of a tangent to a circle at a point $(x_{1}, y_{1})$ on the circle $x^{2} + y^{2} = a^{2}$ .

Figure 5.9. Tanngents on a circle

Let $Q (x_{2}, y_{2})$ be another point on the circle. Then the equation of $P Q$ is $y - y_{1} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} (x - x_{1})$

Since both $P$ and $Q$ lie on the same circle, therefore, $x_{1}^{2} + y_{1}^{2} = a^{2}$ and $x_{2}^{2} + y_{2}^{2} = a^{2}$

Subtracting gives us $\frac{y_{1} - y - 2}{x_{1} - x_{2}} = - \frac{x_{1} + x_{2}}{y_{1} + y_{2}}$

Substituting this in the equation of $P Q$ gives us $y - y_{1} = - \frac{x_{1} + x_{2}}{y_{1} + y_{2}} (x - x_{1})$

As Q $\to P, x_{2} \to x_{1}, y_{2} \to y_{1}$ , and thus,

$y - y_{1} = - \frac{x_{1}}{y_{1}} (x - x - 1) \Rightarrow x x_{1} + y y_{1} = x_{1}^{2} + y_{1}^{2}$ . Thus

x x_{1} + y y_{1} = a^{2}

(5.13)

is the equation of tangent at $P$ on the circle.

Aliter: We know that the tangent to a circle at a point in perpendicular to the radius through that point. The center of our circle is at the origin. Thus, slope of the radius is $\frac{y_{1}}{x_{1}}$ .

So the slope of the tangent will be $- \frac{x_{1}}{y_{1}}$ . Thus, equation of tangent will be $y - y_{1} = - \frac{x_{1}}{y_{1}} (x - x_{1}) \Rightarrow x x_{1} + y y_{1} = a^{2}$ .

Aliter(Using Calculus): The equation of our circle is $x^{2} + y^{2} = a^{2}$ .

Differentiating w.r.t. 𝑥 gives us $2 x + 2 y \frac{d y}{d x} = 0 \Rightarrow \frac{d y}{d x} = - \frac{x}{y}$

Thus, slope at point $P (x_{1}, y_{1})$ is $\frac{d y}{d x} = - \frac{x_{1}}{y_{1}}$ . Thus, equation of tangent will be $y - y_{1} = - \frac{x_{1}}{y_{1}} (x - x_{1}) \Rightarrow x x_{1} + y y_{1} = a^{2}$ .

Now we will find tanegent to the circle represented by the equation $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ .

Like the first method we see that both $P (x_{1}, y_{1})$ and $Q (x_{2}, y_{2})$ lie on the circle. Thus, we can write

$x_{1}^{2} + y_{1}^{2} + 2 g x_{1} + 2 f y_{1} + c = 0$ and $x_{2}^{2} + y_{2}^{2} + 2 g x_{2} + 2 f y_{2} + c = 0$

Subtracting $\frac{y_{1} - y_{2}}{x_{1} - x_{2}} = - \frac{x_{1} + x_{2} + 2 g}{y_{1} + y_{2} + 2 f}$

We know that equation of the line $P Q$ is $y - y_{1} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} (x - x_{1})$

Substituting we get $y - y_{1} = - \frac{x_{1} + x_{2} + 2 g}{y_{1} + y_{2} + 2 f} (x - x_{1})$

As Q $\to P, x_{2} \to x_{1}, y_{2} \to y_{1}$ , and thus,

$y - y_{1} = - \frac{x_{1} + g}{y_{1} + f} (x - x -_{1}) \Rightarrow x x_{1} + y y_{1} + g c + f y = x_{1}^{2} + y_{1}^{2} + g x_{1} + f y_{1}$ $x x_{1} + y y_{1} + g (x + x_{1}) + f (y + y_{1}) + c = 0$ (5.1.4)

Aliter: Center of the circle is $(- g, - f)$ and slope of radius is $\frac{y_{1} - (- f)}{x_{1} - (- g)} = \frac{y_{1} + f}{x_{1} + g}$

So slope of tangent would be $- \frac{x_{1} + g}{y_{1} + f}$ . Thus, equation of tangent is $y - y_{1} = - \frac{x_{1} + g}{y_{1} + f} (x - x_{1})$

Proceeding like earlier the equation of tangent is found to be $x x_{1} + y y_{1} + g (x + x_{1}) + f (y + y_{1}) + c = 0$ .

You are encourage to try the third method using calculus for this as well.

Now we will find equation of the normal at $P (x_{1}, y_{1})$ for the circle $x^{2} + y^{2} = a^{2}$ . Normal is defined as perpendicular to tangent and it always passes through the center of the circle.

Figure 5.10. Normals of a circle

We have found the equation of tangent as $x x_{1} + y y_{1} = a^{2}$ , and thus, its slope is $- \frac{x_{1}}{y_{1}}$ , this makes the slope of the normal as $\frac{y_{1}}{x_{1}}$ .

Since the normal pass through $C (0, 0)$ , therefore, its equationis given by $y = \frac{y_{1}}{x_{1}} x$ i.e. $x y_{1} - x_{1} y = 0$ (5.15)

You are implored to prove this by other techniques as well.

Similarly for the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ , the equation of the normal at $(x_{1}, y_{1})$ is found to be $\frac{x - x_{1}}{x_{1} + g} = \frac{y - y_{1}}{y_{1} + f}$ (5.16)

5.7. Condition that a Line Touches a Circle

We will find the condition for a line $y = m x + c$ to touch the circle $x^{2} + y^{2} = a^{2}$ .

Since the line touches the circle, therefore, $x^{2} + (m x + c)^{2} = a^{2} \Rightarrow (1 + m^{2}) x^{2} + 2 m c x + (c^{2} - a^{2}) = 0$ .

Since the line touches at one point the above equation will have a repeated root and the discriminant must be zero for that. Thus,

$4 m^{2} c^{2} - 4 (1 + m^{2}) (c^{2} - a^{2}) = 0 \Rightarrow c = \pm a \sqrt{1 + m^{2}}$ . Thus, the required condition is $c = \pm a \sqrt{1 + m^{2}}$ (5.17)

Thus, general equation of tangent is $y = m x \pm a \sqrt{1 + m^{2}}$ .

The equation of the line is $y = m x + c$ and equation of tangent is $x x_{1} + y y_{1} = a^{2}$ . Since both equations represnet tangents at $(x_{1}, y_{1})$ , therefore, comparing the coefficients $\frac{x_{1}}{- m} = \frac{y_{1}}{1} = \frac{a^{2}}{c} \Rightarrow x_{1} = - \frac{a^{2} m}{c}, y_{1} = \frac{a^{2}}{c}$

Thus, point of contact is $(- \frac{a^{2} m}{c}, \frac{a^{2}}{c})$ .

5.7.1. Two Tangents to a Circle

We will prove that from a point outside the circle two tangents can always be drawn to the circle.

Let the circle be $x^{2} + y^{2} = a^{2}$ . We know that the equation of tangent is $y = m x + a \sqrt{1 + m^{2}}$

Let the external point be $(x_{1}, y_{1})$ . Since the tangent passes through it, therefore, $y_{1} = m x_{1} + a \sqrt{1 + m^{2}} \Rightarrow (y_{1} - m x_{1})^{2} = a^{2} (1 + m^{2})$ , which is a quadratic equation in $m$ . Thus, it will give two values of $m$ , and hence, two different tangents can be drawn from the same point.

If we check the nature of roots then discriminant is given by $x_{1}^{2} + y_{1}^{2} - a^{2}$ i.e. if the point is external then two real 𝑚s mean two different tangent.

5.7.2. Length of a Tangent

We will find the length of a tangent from an external point to a circle.

Let the point be $P (x_{1}, y_{1})$ and the circle be $x^{2} + y^{2} = a^{2}$ . From point $P$ draw a tangent to the circle, whihc touches the circle at $T$ . $C T$ will be perpendicular to $P T$ .

Figure 5.11. Length of a tangent to a circle

Thus, in right-angle $△ C P T$ , we have $P T^{2} = C P^{2} - C T^{2} = x_{1}^{2} + y_{1}^{2} - a^{2}$ . Thus, $P T = \sqrt{x_{1}^{2} + y_{1}^{2} - a^{2}}$ (5.18)

Similarly if the circle is $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ and the center is $(- g, - f)$ then $P T^{2} = (x_{1} + g)^{2} + (y_{1} + f)^{2} - (g^{2} + f^{2} - c)$

P T = \sqrt{x_{1}^{2} + y_{1}^{2} + 2 g x_{1} + 2 f y_{1} + c}

(5.19)

5.7.3. Pair of Tangents

We will find the equation of pair of tangents drawn from an external point to a circle.

Figure 5.12. Pair of a tangent to a circle

Let the point be $P (x_{1}, y_{1})$ from which we draw two tangents $P A$ and $P B$ to the circle, touching the circle at $A$ and $B$ respectively.

Let $R (α, β)$ be any point on any one of the tangents, say $P A$ . Then the locus of $R (α, β)$ will be the required equation of the pair of tangents to the circle from point $(x_{1}, y_{1})$ .

Equation of $P R$ is $y - y_{1} = \frac{β - y_{1}}{α - x_{1}} (x - x_{1}) \Rightarrow y (α - x_{1}) - x (β - y_{1}) - α y_{1} + β x_{1} = 0$

Now $A C = a \Rightarrow \frac{| β x_{1} - α y_{1} |}{\sqrt{(α - x_{1})^{2} + (β - y_{1})^{2}}} = a \Rightarrow \frac{(β x_{1} - α y_{1})^{2}}{(α - x_{1})^{2} + (β - y_{1})^{2}} = a^{2}$

So locus of $R (α, β)$ is $(y x_{1} - x y_{1})^{2} = a^{2} [(x - x_{1})^{2} + (y - y_{1})^{2}]$

(x_{1}^{2} + y_{1}^{2} - a^{2}) (x^{2} + y^{2} - a^{2}) = {(x x_{1} + y y_{1} - a^{2})}^{2}

(5.20)

This is the required equation of pair of tangents drawn from $(x_{1}, y_{1})$ . If circle is denoted by $S$ then the pair of tangents is given by $S S_{1} = T^{2}$ .

For the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ the pair of tangents is given by $\begin{array}{r} (x_{1}^{2} + y_{1}^{2} + 2 g x_{1} + 2 f y_{1} + c) (x^{2} + y^{2} + 2 g x + 2 f y + c) = \\ {[x x_{1} + y y_{1} + g (x + x_{1}) + f (y + y_{1}) + c]}^{2} \end{array}$ (5.21)

5.8. Chord of Contact of Tangents

Figure 5.13. Chord of contact of tangents to a circle

Let $P$ be a point outside a circle. From $P$ two tangents $P A$ and $P B$ can be drawn to touch the circle at $A$ and $B$ , respectively just like last section. The chord $A B$ is called the chord of contact for point $P$ .

We will now find the equation of this chord of contact.

Let $A = (x_{2}, y_{2})$ and $B = (x_{3}, y_{3})$ as shown in the figure.

Since these points lie on the tangents, therefore, $x x_{2} + y y_{2} = a^{2}$ and $x x_{3} + y y_{3} = a^{2}$ .

Since both pass through $(x_{1}, y_{1})$ , therefore, $x_{1} x_{2} + y_{1} y_{2} = a^{2}$ and $x_{1} x_{3} + y_{1} y_{3} = a^{2}$ .

Thus, we can say that $x x_{1} + y y_{1} = a^{2}$ passes through $A$ and $B$ . Hence, the equation of line $A B$ is this equation.

Similarly, for the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ the equation of chord of contact is given by $x x_{1} + y y_{1} + g (x + x_{1}) + f (y + y_{1}) + c = 0$ .

5.9. Poles and Polars

If from a point $P$ any straight line is drawn to meet the circle in $Q$ and $R$ and tangents at $Q$ and $R$ meet at $T$ then locus of $T$ is called the polar of $P$ w.r.t to the circle and $P$ is called the pole of its polar i.e. $T$ .

Figure 5.14. Pole and polar of a circle

The point $P$ can be inside or outside the circle. The diagram shows one such $P$ inside the circle through which the line $Q R$ passes and tangents meets at $T (α, β)$ .

Clearly, $Q R$ is the chord of contact of $T$ whose equation is $x α + y β = a^{2}$ , but it passes through $P (x_{1}, y_{1})$ , therefore, $x_{1} α + y_{1} β = a^{2}$

Putting $(x, y)$ instead of $(α, β)$ we get the locus of $T$ as $x x_{1} + y y_{1} = a^{2}$ (5.22)

This is the required equation of the polar of point $P (x_{1}, y_{1})$ and as can be seen from the equation it is a straight line.

Similarly, for circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ the equation of the polar is given by $x x_{1} + y y_{1} + g (x + x_{1}) + f (y + y_{1}) + c = 0$ (5.23)

5.9.1. Coordinates of a Pole

Now we will find coordinates of pole of a line.

Consider a line $l x + m y + n = 0$ whose pole is to be found w.r.t. the circle $x^{2} + y^{2} = a^{2}$ .

Let the pole be $(x_{1}, y_{1})$ . Now the equation of polar of point $(x_{1}, y_{1})$ w.r.t. to the circle is given by $x x_{1} + y y_{1} = a^{2}$ .

Comapring coefficients with the given line $\frac{l}{x_{1}} = \frac{m}{y_{1}} = - \frac{n}{a^{2}} \Rightarrow x_{1} = - \frac{a^{2} l}{n}, y_{1} = - \frac{a^{2} m}{n}$

Hence, coordinate of the pole is given by $(- \frac{a^{2} l}{n}, - \frac{a^{2} m}{n})$ .

5.9.2. Properties of Poles and Polars

If the polar of a point $P$ w.r.t. to a circle passes through $Q$ then the polar of $Q$ w.r.t. the circle will pass through $P$ .
Let the equation of the circle is $x^{2} + y^{2} = a^{2}$ . Also, let $P = (x_{1}, y_{1})$ and $Q = (x_{2}, y_{2})$ .
Now equations of the polars of these two points will be $x x_{1} + y y_{1} = a^{2}$ and $x x_{2} + y y_{2} = a^{2}$ .
If the polar of $P$ passes through $Q$ then $x_{1} x_{2} + y_{1} y_{2} = a^{2}$ .
Thus, we have proven our assumption. The points $P$ and $Q$ are called conjugate points.
f the pole of a line w.r.t. a circle lies on another line, then the pole of the other line w.r.t. the same circle will lie on the first line.
Let the circle be $x^{2} + y^{2} = a^{2}$ and the lines be $l x + m y + n = 0$ and $l^{'} x + m^{'} y + n^{'} = 0$ .
Let the pole of first line w.r.t to circle be $(x_{1}, y_{1})$ . Then the polar will be $x x_{1} + y y_{1} = a^{2}$ . Comapring coefficients we arrive at $(x_{1}, y_{1}) = (- \frac{a^{2} l}{n}, - \frac{a^{2} m}{n})$ .
This point will lie on the second line, therefore, $- \frac{a^{2} l l^{'}}{n} - \frac{a^{2} m m^{'}}{n} + n^{'} = 0 \Rightarrow a^{2} l l^{'} + a^{2} m m^{'} - n n^{'} = 0$ .
Similarly we let pole of the second line w.r.t. to the circle be $(x_{2}, y_{2})$ . Then the polar will be $x x_{2} + y y_{2} = a^{2}$ . Comapring coefficients we arrive at $(x_{2}, y_{2}) = (- \frac{a^{2} l^{'}}{n^{'}}, - \frac{a^{2} m^{'}}{n^{'}})$
This point will lie on the first line, therefore, $a^{2} l l^{'} + a^{2} m m^{'} - n n^{'} = 0$ .
Thus, the points lies on corresponding lines. Such lines are called conjugate lines.
If the polars of two point $P$ and $Q$ w.r.t. a circle meet at $R$ , then $R$ is the pole of the line $P Q$ .
Let $P (x_{1}, y_{1})$ and $Q (x_{2}, y_{2})$ be two points. Then the polar $A B$ and $C D$ of $P$ and $Q$ w.r.t. to the circle $x^{2} + y^{2} = a^{2}$ will be $x x_{1} + y y_{1} = a^{2}$ and $x x_{2} + y y_{2} = a^{2}$ .
According to question they meet at $R$ . Solving the two equations, we have $\frac{x}{y_{1} - y_{2}} = \frac{y}{x_{2} - x_{1}} = - \frac{a^{2}}{x_{1} y_{2} - x_{2} y_{1}}$ , which gives $R$ .
We have to prove that polar of $R$ w.r.t. the circle is line $P Q$ . The polar of $R$ w.r.t. the circle is $x (- \frac{a^{2} (y_{1} - y_{2})}{x_{1} y_{2} - x_{2} y_{1}}) + y (- \frac{a^{2} (x_{1} - x_{2})}{x_{1} y_{2} - x_{2} y_{1}}) = a^{2}$
$\Rightarrow \frac{y - y_{1}}{y_{1} - y_{2}} = \frac{x - x_{1}}{x_{1} - x_{2}}$ , which is the line $P Q$ .

5.10. Equation of a Chord

We will find equation of a chord whose midpoint is given.

Let the equation of the circle be $x^{2} + y^{2} = a^{2}$ with center $C (0, 0)$ .

Let $P Q$ be the chord whose mid-point is $L (x_{1}, y_{1})$ . Slope of line $C L = - \frac{x_{1}}{y_{1}}$

Since $P Q ⊥ C L$ , herefore, slope of $P Q = \frac{y_{1}}{x_{1}}$ . Equation of $P Q$ is $y - y_{1} = \frac{y_{1}}{x_{1}} (x - x_{1})$

Figure 5.15. Chord of a circle

Thus, equation of the chord is given by $x x_{1} + y y_{1} = x_{1}^{2} + y_{1}^{2}$ (5.24) Similarly the equation of chord for the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ is given by $x x_{1} + y y_{1} + g (x + x_{1}) + f (y + y_{1}) + c = x_{1}^{2} + y_{1}^{2} + 2 g x_{1} + 2 f y_{1} + c$ (5.25)

5.11. Intersection of Circles

The angle between two circles is the angle between their tangents at their point of intersection. Let $P$ be a point of intersection of two circles. Let $P T$ and $P T^{'}$ be the tangents to the two circles at the point of intersection. Then the angle is defined at $∠ T P T^{'}$ or $π - ∠ T P T^{'}$ . If $C_{1}$ and $C_{2}$ be the centers of the two circles then

$P C_{1} ⊥ P T$ and $P C_{2} ⊥ P T^{'} ∴ ∠ T P T^{'} = ∠ C_{1} P C_{2}$ or $π - C_{1} P C_{2}$ .

Figure 5.16. Intersecrtion of circles

5.11.1. Orthogonal Circles

Two circles are said to intersect orthogonally if they intersect at right angles. We will find the condition for two circles $x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y + c_{1} = 0$ and $x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y + c_{2} = 0$ are orthogonal.

Let $A$ and $B$ be the center of these circles and $r_{1}$ and $r_{2}$ be their radii respectively. Then

Figure 5.17. Orthogonal circles

Since the circles are orthogonal $P C_{1}^{2} + P C_{2}^{2} = C_{1} C_{2}^{2} \Rightarrow r_{1}^{2} + r_{2}^{2} = C_{1} C_{2}^{2}$

$\Rightarrow g_{1}^{2} + f_{1}^{2} - c_{1} + g_{2}^{2} + f_{2}^{2} - c_{2} = (g_{2} - g_{1})^{2} + (f_{2} - f_{1})^{2}$ $2 g_{1} g_{2} + 2 f_{1} f_{2} = c_{1} + c_{2}$ (5.26)

5.12. Radical Axis

Figure 5.18. Radical axis of two circles

The radical axis of two circles is the locus of a point which moves so that the length of the tangents drawn from it to the two points are equal.

We will find equation of radical axis of two circles $x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y + c_{1} = 0$ and $x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y + c_{2} = 0$ .

Let $P (α, β)$ be any point from which the tangents from it to these circles are equal. Let the tangents from this point to the two circles be $P A$ and $P B$ .

Now $P A = \sqrt{α^{2} + β^{2} + 2 g_{1} α + 2 f_{1} β + c_{1}}$ and $P B = \sqrt{α^{2} + β^{2} + 2 g_{2} α + 2 f_{2} β + c_{2}}$

Figure 5.19. Radical axis of two circles

We have $P A = P B \Rightarrow 2 (g_{1} - g_{2}) α + 2 (f_{1} - f_{2}) β + c_{1} - c_{2} = 0$

Hence, locus of $P$ is $2 (g_{1} - g_{2}) x + 2 (f_{1} - f_{2}) y + c_{1} - c_{2} = 0$ (5.27) which is a straight line and equation of radical axis.

5.12.1. Properties of Radical Axes

The radical axis of two circles is perpendicular to the line of centers.
Let the two circles be $x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y + c_{1} = 0$ and $x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y + c_{2} = 0$ .
Let $A$ and $B$ be the centers of these two circles. Then slope of $A B$ is $\frac{f_{2} - f_{1}}{g_{2} - g_{1}}$ , which is negative reciprocal of $2 (g_{1} - g_{2}) x + 2 (f_{1} - f_{2}) y + c_{1} - c_{2} = 0$ .
Thus, the radical axis of two circles is perpendicular to the line of centers.
The radical axis of three circles takes two at a time are concurrent.
Consider three circles, which are $x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y + c_{1} = 0, x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y + c_{2} = 0$ and $x^{2} + y^{2} + 2 g_{3} x + 2 f_{3} y + c_{3} = 0$ .
So radical axes will be $2 (g_{1} - g_{2}) x + 2 (f_{1} - f_{2}) y + c_{1} - c_{2} = 0, 2 (g_{2} - g_{3}) x + 2 (f_{2} - f_{3}) y + c_{2} - c_{3} = 0$ and $2 (g_{3} - g_{1}) x + 2 (f_{3} - f_{1}) y + c_{3} - c_{1} = 0$ .
Adding these we get equality, hence, the three lines are concurrent. The point where these radical axes meet is called the radical center of the three circles.
The radical axis of the two circles bisect their common tangent.
Let $A B$ be one of the common tangents meeting the raidcal axis at $P$ , then since $P$ lies on the radical axis, hence, by definition of radical axis $P A = P B$ .
Thus, the radical axis bisect the common tangent.
he locus of the cetner of a circle cutting two given circles orthogonally is the radical axis of the two circles.
Consider three circles, which are $x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y + c_{1} = 0, x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y + c_{2} = 0$ and $x^{2} + y^{2} + 2 g_{3} x + 2 f_{3} y + c_{3} = 0$ . Let them cut each other orthogonally.
$2 g_{1} g_{2} + 2 f_{1} f_{2} = c_{1} + c_{2}$ and $2 g_{1} g_{3} + 2 f_{1} f_{3} = c_{1} + c_{3}$
Subtracting we get $2 g_{1} (g_{2} - g_{3}) + 2 f_{1} (f_{2} - f_{3}) = c_{2} - c_{3}$
Thus, locus of the center is $2 (g_{1} - g_{2}) x + 2 (f_{1} - f_{2}) y + c_{1} - c_{2} = 0$ which is the radical axis of the two circles being cut orthogonally.

5.13. Coaxial Circles

A system of circles is said to be coaxial if each pair of circles of the system has the same radical axis.

Consider two circles $x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y + c_{1} = 0$ and $x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y + c_{2} = 0$ .

Equation of radical axis will be $2 (f_{1} - f_{2}) x + 2 (f_{1} - f_{2}) y + c_{1} - c_{2} = 0$ .

We know that the radical axis of two circles is perpendicular to the line joining their centers, therefore, if we take the line joining the centers as $x$ -axis and radical axis as $y$ -axis, then

$f_{1} = f_{2} = 0$ and equation of radical axis will be $c_{1} - c_{2} = 0 \Rightarrow c_{1} = c_{2}$ .

Thus, constant term in all the circles must be same for them to be coaxial circles.

5.13.1. Limiting Points of a Coaxial System

Consider a system of coaxial circles having equation $x^{2} + y^{2} + 2 g x + c = 0$ , where $c$ is a constant and $g$ is a parameter.

The radius is $\sqrt{g^{2} - c}$ . If $g = \pm \sqrt{x}$ then the radius vanishes and the circle becomes point circle.

Thus, there are two circles of the system whose radii are zero, centers of these circles are $(\sqrt{c}, 0)$ and $(- \sqrt{c}, 0)$ .

These points are called limiting points of the system.

5.13.2. Circles through the Points of Intersection of a Circle and a Line

Consider a circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ and a line $l x + m y + n = 0$ .

Consider the equation $x^{2} + y^{2} + 2 g x + 2 f y + c + λ (l x + m y + n) = 0$ which is also the equation of a circle.

The coordinates which satisfy the considered circle and line also satisfy this equation of circles. Thus, point of intersection of the considered circle and line lies on this new circle.

Similarly we can say that $x^{2} + y^{2} + 2 g x + 2 f y + c + λ (x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y + c) = 0$ is the locus of points of intersection of two circles.

5.14. Problems

Find the center and the radius of the circle $3 x^{2} + 3 y^{2} - 8 x - 10 y + 3 = 0$ .
Prove that the radii of the circles $x^{2} + y^{2} = 1, x^{2} + y^{2} - 2 x - 6 y = 6$ and $x^{2} + y^{2} - 4 x - 12 y = 9$ are in A.P.
Find the area of an equilateral triangle inscribed in the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ .
Find the center and the radius of the circle $x^{2} + y^{2} + 6 x - 8 y = 24$ .
Find the center and the radius of the circle $5 x^{2} + 5 y^{2} + 4 x - 8 y = 16$ .
Find the center and the radius of the circle $x^{2} + y^{2} - 6 x - 2 y = 6$ .
What will be the radius and center of the circle $\frac{1}{2} (x^{2} + y^{2}) + x \cos θ + y \sin θ - 4 = 0$ ?
Prove that the centers of the circles $x^{2} + y^{2} = 1, x^{2} + y^{2} + 6 x - 2 y - 1 = 0$ and $x^{2} + y^{2} - 12 x + 4 y = 1$ are collinear.
Prove that the radii of the circles $x^{2} + y^{2} = 1, x^{2} + y^{2} - 2 x - 6 y = 6$ and $x^{2} + y^{2} - 4 x - 12 y = 9$ are in A.P.
Prove that the radii of the circles $x^{2} + y^{2} = 4, 4 x^{2} + 4 y^{2} - 8 x - 24 y + 15 = 0$ and $x^{2} + y^{2} - 4 y - 5 = 0$ are in A.P.
Prove that the circles $x^{2} + y^{2} - 9 = 4 r^{2}, r = 1, 2, 3$ cut off equal intercepts between the circles on the line $3 x + 4 y + 15 = 0$ .
Find the equation of the circle whose center is $(1, 2)$ and which passes through the point $(4, 6)$ .
If the equations of two diameters of a circle are $x + y = 6$ and $x + 2 y = 4$ and the radius of the circle is $10$ , find the equation of the circle.
Find the equation of the circle whose center is $(3, 4)$ and which touches the line $5 x + 12 y = 1$ .
Find the equation of a circle which passes through the point $(2, 0)$ and whose center is the limit of the point of intersection of the lines $3 𝑥 + 5 𝑦 = 1$ and $(2 + c) x + 5 c^{2} y = 1$ as $c$ tends to $1$ .
A circle has radius of $3$ units and its center lies on the line $y = x - 1$ . Find the equation of the circle if it passes through $(7, 3)$ .
Find the equation of the circle, which touches the axes and whose center lies on the line $x - 2 y = 3$ .
A circle of radius $2$ lies in the first quadrant and touches both the axes of coordinates. Find the equation of the circle with center at $(6, 5)$ and touching the above circle externally.
Find the equation of the circles whose center is $(3, - 1)$ and which cut off an intercept of length 6 from $2 x - 5 y + 18 = 0$ .
Find equations of the circles touching $y$ -axis at $(0, 3)$ and making intercept of $8$ units on the $x$ -axis.
Find the equation of the circle having the pair of lines $x^{2} + 2 x y + 3 x + 6 y = 0$ as its normals and having the size just sufficient to contain the circle $x (x - 4) + y u y (y - 3) = 0$ .
A circle of raidus $5$ units touches the coordinate axes in the first quadrant. If the circle makes one complete roll on the $x$ -axis along the positive direction of $x$ -axis, find its equation in the new position.
The circle $x^{2} + y^{2} - 4 x - 8 y + 16 = 0$ rolls up the tangent to it at $(2 + \sqrt{3}, 3)$ by $2$ units, assuming the $x$ -axis as horizontal, find the equation of the circle in the new position.
Find the equations of the circles touching the lines $y = 0$ and $y = \sqrt{3} (𝑥 + 1)$ and having the centers at a distance $1$ from the origin.
Find the equation of the circle passing through the point $(2, 8)$ touching the lines $4 x - 3 y + 24 = 0$ and $4 x + 3 y - 42 = 0$ and having $x$ coordinate of the center of the circle numerically less than or equal to $8$ .
Find the equation of the circle whose center is $(1, - 5)$ and radius is $7$ .
Find the equation of the circle whose center is $(- 1, - 2)$ and diameter is $25$ .
If the equation of two diameters of a circle are $2 x + y = 6$ and $3 x + 2 y = 4$ and the radius is $10$ , find the equation of the circle.
Find the equation of the circle which passes through the point of intersection of $3 x - 2 y - 1 = 0$ and $4 x + y - 27 = 0$ and whose center is $(2, 3)$ .
Find the equation of the circle whose center is $(1, 2)$ and which passes through the point of intersection of $3 x + y = 14$ and $2 x + 5 y = 18$ .
ind the equation of the circle passing through the center of the circle $x^{2} + y^{2} - 4 x - 6 y = 8$ and being concentric with the circle $x^{2} + y^{2} - 2 x - 8 y = 5$ .
Find the equation of the circle passing through the point of intersection of $x + 3 y = 0$ and $2 x - 7 y = 0$ and whose center is the point of intersection of lines $x + y + 1 = 0$ and $x - 2 y + 4 = 0$ .
Find the equation of the circle whose radius is $5$ and the center lies on the positive side of $x$ -axis at a distance $5$ from the origin.
Find the equation of the circle which passes through the points $(- 1, 2)$ and $(3, - 2)$ and whose center lies on the line $x - 2 y = 0$ .
Find the equation of the circle which passes through the point $(1, - 2)$ and $(4, - 3)$ and whose center lies on the line $3 x + 4 y = 7$ .
Find the equation of the circle whose radius is $5$ and which touches the circle $x^{2} + y^{2} - 2 x - 4 y - 20 = 0$ at the point $(5, 5)$ .
Find the equation of the circle whose center is $(1, - 3)$ and which touches the line $2 x - y - 4 = 0$ .
Write down the equation of a circle concentric with the circle $x^{2} + y^{2} - 4 x + 6 y - 17 = 0$ and tangent to the line $3 x - 4 y + 7 = 0$ .
Find the equation of the circle of radius $5$ and touching the line $3 x - 4 y + 5 = 0$ at $(1, 2)$ .
If the raidus of the circle is 5 and the equations of the twp normals to the circle are $3 x - 5 y + 2 = 0$ and $x + 2 y = 3$ , find the equation of the circle.
Find the equation of the circle which touches the $y$ -axis at a distance $4$ from the origin towards the positive side of $y$ -axis and cuts of an intercept $6$ on the $x$ -axis.
Find the equation of the circle which touches both the axes and whose radius is $a$ .
Find the equation of the circle passing through the point $(h, k)$ and touching the $y$ -axis at origin.
Find the equation of the circle touching the axis of $x$ at the origin and touching the line $3 y = 4 x + 24$ .
What is the parametric equation of the circle $x^{2} + y^{2} = 16$ .
Find the equation of the circle which touches the line $2 x - y = 1$ at $(1, 1)$ and the line $2 x + y = 4$ .
Find the euqation of the circle touching the lines $4 x - 3 y = 30$ and $4 x - 3 y + 10 = 0$ having the center on the line $2 x + y = 0$ .
A circle of raidus $4$ units touches the coordinate axes in the first quadrant. Find the equation of its image w.r.t. the line mirror $y = 0$ .
The equation of a circle is $x^{2} + y^{2} + 16 x - 24 y + 183 = 0$ . Find the equation of the image of this circle by the line mirror $4 x + 7 y + 13 = 0$ .
The circle $(x - a)^{2} + (y - a) 2 = a^{2}$ is rolled on the 𝑥-axis in the positive direction through one complete revolution. Find the equation of the circle in the new position.
The center of a circle is $(1, 1)$ and its radius is $5$ units. If the center is shifted on the line $x - y = 0$ through a distance of $\sqrt{2}$ units, find the equation of the circle in the new position. How many such circles are possible?
Find the equation of the circles which pass through the origin and cut off equal chords of $\sqrt{2}$ units from the straight lines $y = x$ and $y = - x$ .
Find the center of the circle which is inscribed in the triangle formed by the lines $3 x + 4 y - 15 = 0, 3 x - 4 y - 7 = 0$ and $12 x + 5 y - 115 = 0$ .
Show that the equation of the circle which touches the coordinate axes and whose center lies on the straight line $l x + m y + n = 0$ is $(l \pm m)^{2} (x 2 + y^{2}) \pm 2 n (l \pm m) (x + y) + n^{2} = 0$ .
Of the two concentric circles the smaller one has the equation $x^{2} + y^{2} = 4$ . If the intercept on the line $x + y = 2$ made between the two circles is $1$ , find the equation of the larger circle.
Is $(3, 2)$ an interior point or an exterior point of the circle $x^{2} + y^{2} - 2 x + y = 0$ ? If inerior, find the equation of the circle centered on it and of maximum area contained in the given in the circle. If exterior, find the circle of maximum radius centered on it containing the given circle.
Find the equation of the circle with minimum raidus which contains the three circles $x^{2} + y^{2} - 4 y - 5 = 0, x^{2} + y^{2} + 12 x + 4 y + 31 = 0$ and $x^{2} + y^{2} + 6 x + 12 y + 36 = 0$ .
Find the equation of the circle whose diameter is the line joining the points $(- 4, 3)$ and $(12, - 1)$ . Find also the intercept made by it on $y$ -axis.
The sides of a square are $x = 1, x = 3, y = 2$ and $y = 4$ . Find the equation of the circle drawn on the diagonals of the square as its diameter.
Find the equation of the circle on the line joining the origin and $(2, - 4)$ as diameter.
Find the equation of the circle, the endpoints of whose diameter are $(2, - 3)$ and $(- 2, 4)$ . Find its center and radius.
Find the equation of the circle drawn on the intercept between the axes made by the line $3 x + 4 y = 12$ as a diameter.
Find the equation of the circle the endpoints of whose diameter are the centers of the circles $x^{2} + y^{2} + 6 x - 14 y = 1$ and $x^{2} + y^{2} - 4 x + 1 - y = 2$ .
The sides of a square are $x = 6, x = 9, y = 3$ and $y = 6$ . Find the equation of a circle drawn on the diagonal of the square as its diameter.
Find the equation of the circle circumscribing the rectangle whose sides are $x - 3 y = 4, 3 x + y = 22, x - 3 y = 14$ and $3 x + y = 62$ .
The abscissa if two points $A$ and $B$ are the roots of the equation $x^{2} 2 + 2 x - a^{2} = 0$ and the oridinates are the roots of the equation $y^{2} + 4 y - b^{2} = 0$ . Find the equation of the circle with $A B$ as its diameter. Also find the coordinates of the center and the length of the raidus of the circle.
If $(4, 1)$ be one endpoint of the diameter of the circle $x^{2} 2 + y^{2} - 2 x + 6 y - 15 = 0$ , find the coordinates of the other endpoint of the diagonal.
ind the equation of the circumcircle of the quadrilateral formed by the four lines $a x + b y \pm c = 0$ and $b x - a y \pm c = 0$ .
Find the equation of the circle which passes through the points $(1, - 2)$ and $(4, - 3)$ and whose center lies on the line $3 x + 4 y = 7$ .
Find the equation of the circle passing through the origin and the points where the straight line $3 x + 4 y = 12$ meets the coordinate axes.
Show that the cyclic quadrilateral is formed by the lines $5 x + 3 y = 9, x = 3 y, 2 x = y$ and $x + 4 y + 2 = 0$ taken in order. Find the equation of the circle.
Find the equation of a circle passing through the points $(1, 2)$ and $(3, 4)$ and touching the line $3 x + y - 3 = 0$ .
A circle touches both the $x$ -axis and the line $4 x - 3 y + 4 = 0$ . Its center is in the third quadrant anf lies on the line $x - y - 1 = 0$ . Find the equation of the circle.
Find the equation of the circle passing through the points $(1, 0), (0, 1)$ and $(1, - 2)$ .
Find the equation of the circle passing through the points $(0, 0), (a, 0)$ and $(0, b)$ .
Find the equation of the circle which passes through the origin and cuts off chords of length $4$ and $6$ on the positive side of $x$ -axis and $y$ -axis respectively.
Find the equation of the circumcircle of the triangle formed by the lines $y = x, y = 2 x$ and $y = 3 x + 2$ .
Find the incenter of the triangle whose sides are $7 x - y + 11 = 0, x + y - 15 = 0$ and $7 x + 17 y + 65 = 0$ . Find the equation of the incircle.
Find the equation of the circles passing through the origin and cutting off equal intercepts $1$ from the lines $3 x = 4 y$ and $4 x = 3 y$ .
Find the equation of the circle described on the common chord of the circles $x^{2} + y^{2} - 4 x - 5 = 0$ and $x^{2} + y^{2} + 8 y + 7 = 0$ as diameter.
Show that the circle on the chord $x \cos α + y \sin α - p = 0$ of the circle $x^{2} + y^{2} = a^{2}$ as diameter is $x^{2} + y^{2} - a^{2} - 2 p (x \cos α + y \sin α - p) = 0$ .
Prove that the equation $x^{2} + y^{2} - 4 + k (y - m x - 2 \sqrt{1 + m^{2}}) = 0$ represents a family of circles touching each other at a common point for all $k$ , where $m$ is a given constant.
Show that the general equation of a circle which passes through the points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ may be written as $(x - x_{1}) (x - x_{2}) + (y - y_{1}) (y - y_{2}) + λ | \begin{matrix} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{matrix} | = 0$ , and hence, deduce the diameter form of the equation of a circle.
The line $A x + B y + C = 0$ cuts the circle $x^{2} + y^{2} + a x + b b y + c = 0$ in $P$ and $Q$ . The line $A^{'} x + B^{'} y + C^{'} = 0$ cuts the circle $x^{2} 2 + y^{2} + a^{'} x + b^{'} y + c^{'} = 0$ in $R$ and $S$ . If $P, Q, R, S$ are concyclic points show that $| \begin{matrix} a - a^{'} & b - b^{'} & c - c^{'} \\ A & B & C \\ A^{'} & B^{'} & C^{'} \end{matrix} | = 0$ .
A fixed circle is cut by circles passing through two given points $A (x_{1}, y_{1})$ and $B (x_{2}, y_{2})$ . Show that the chord of intersection of the fixed circle with any one of the circled, passes through a fixed point.
Tangents $P Q$ and $P R$ are drawn to the circle $x^{2} + y^{2} = a^{2}$ from the point $P (x_{1}, y_{1})$ . Find the equation of the circumcircle of $△ P Q R$ .
Find the equation of the circle passing through the point of intersection of the circles $x^{2} + y^{2} - 6 x + 2 y + 4 = 0$ and $x 2 + y^{2} + 2 x - 4 y - 6 = 0$ and with its center on the line $y = x$ .
If the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ bisects the circumference of the circle $x^{2} + y^{2} + 2 g^{'} x + 2 f^{'} y + c^{'} = 0$ , prove that $2 g^{'} (g - g^{'}) + 2 f^{'} (f - f^{'}) = c - c^{'}$ .
Find the equation of the circle of radius 4 and passing through the point of intersection of circles $x^{2} + y^{2} - 2 x - 4 y - 4 = 0$ and $x^{2} + y^{2} - 10 x - 12 y + 40 = 0$ .
Show that the common chord of the circles $x^{2} + y^{2} - 6 x - 4 y + 9 = 0$ and $x^{2} + y^{2} - 8 x - 6 y + 23 = 0$ pass through the center of the second circle and find its length.
Find the equation of the circle whose diameter is the common chord of the circles $x^{2} + y^{2} + 2 x + 3 y + 1 = 0$ and $x^{2} + y^{2} + 4 x + 3 y + 2 = 0$ .
If $y = m x$ be the equation of a chord of the circle $x^{2} + y^{2} - 2 a x = 0$ , prove that the circle of which this chord is a diameter has the equation $(1 + m^{2}) (x^{2} + y^{2}) - 2 a (x + m y) = 0$ .
Find the equation of the circle which passes through the point of intersection of the circles $x^{2} + y^{2} - 6 x + 2 y + 4 = 0$ and $x^{2} + y^{2} + 2 x - 4 y - 6 = 0$ and whose center lies on the line $y = x$ .
Prove that the equation $x^{2} + y^{2} + 2 (3 + p) x + 2 (3 - p) y + 4 = 0$ represents a circle for all values of $p$ , passing through two fixed points. Find the fixed points.
Find the equation of the circle through the points of intersection of the circles $x^{2} + y^{2} = 4 a^{2}$ and $x^{2} + y^{2} - 2 x - 4 y + 4 = 0$ and touching the line $x + 2 y = 0$ .
Find the equation of the circle which passes through the points of intersection of the circle $x^{2} + y^{2} - x - y = 0$ and the line $x + y = 1$ and also through the point $(1, 1)$ .
Find the equation of the circle which has for its diameter the chord cut off on the line $p x + q y - 1 = 0$ by the circle $x^{2} + y^{2} = 𝑎^{2}$ .
The point $A (α, β)$ is outside the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ and $A P, A Q$ are tangents to the circle. Find the equation of the circumcircle of $△ P Q R$ .
Show that the line $3 x - 4 y = c$ will meet the circle having center at $(2, 4)$ and the radius $(2, 4)$ in real and distinct points if $- 35 < c < 15$ .
Find the length of the chord $4 x - 3 y = 5$ of the circle $x^{2} + y^{2} + 3 x - y - 10 = 0$ .
Prove that the line $y = x + a \sqrt{2}$ touches the circle $x^{2} + y^{2} = a^{2}$ . Also find the point of contact.
Find the equation of tangents of the circle $x^{2} + y^{2} - 6 x + 4 y - 12 = 0$ which are parallel to the line $4 x + 3 y + 5 = 0$ .
Show that the common tangents to the circles $x^{2} + y^{2} - 6 x = 0$ and $x^{2} + y^{2} + 2 x = 0$ form an equilateral triangle.
Three concentric circles of which the biggest is $x^{2} + y^{2} = 1$ have their radii in A.P. If the line $y = x + 1$ cuts all the circles in real and distinct points then find the interval in which the common difference of the A.P. will lie.
If $4 l^{2} - 5 m^{2} + 6 l + 1 = 0$ , then shown that the line $l x + m y + 1 = 0$ touches a fixed circle. Find the center and raidus of the circle.
Find the point $P$ on the circle $x^{2} + y^{2} - 4 x - 6 y + 9 = 0$ such that (i) $∠ P O X$ is minimum (ii) $O P$ is maximum, where $O$ is the origin and $O X$ is the $x$ -axis.
Show that the circle $x^{2} + y^{2} - 2 a x - 2 a y + a^{2} = 0$ touches both the axes. Also, find the point of contact.
Find the length of the chord of the circle $x^{2} + y^{2} = 16$ which bisects the join of the points $(2, 3)$ and $(1, 2)$ perpendicularly.
Find the length of the chord intercepted by the straight line $x - 7 y + 4 = 0$ and the circle $x^{2} + y^{2} - 14 x + 4 y + 28 = 0$ . Also, find the middle point of the chord.
Find the length of the common chords of the circles $x^{2} + y^{2} + 3 x + 5 y + 4 = 0$ and $x^{2} + y^{2} + 5 x + 3 y + 4 = 0$ .
Find the equation and length of the common chord of the circles $x^{2} + y^{2} + 2 x + 3 y + 1 = 0$ and $x^{2} + y^{2} + 4 x + 3 y + 2 = 0$ .
Prove that the length of the common chord of the circlea $(x - a)^{2} + (y - b)^{2} = c^{2}$ and $(c - b) 2 + (y - a)^{2} = c^{2}$ is $\sqrt{4 c^{2} - 2 (a - b)^{2}}$ . Hence find the condition that the two circles may touch each other.
Prove that the length of the common chord of the two circles $x^{2} + y^{2} + 2 h x + a^{2} = 0$ and $x^{2} + y^{2} - 2 k y - a^{2} = 0$ is $2 \sqrt{\frac{(h^{2} - a^{2}) (k^{2} + a^{2})}{h^{2} + k^{2}}}$ .
Prove that the length of the common chord of the circles $x^{2} + y^{2} + a x + b y + c = 0$ and $x^{2} + y^{2} + b x + a y + c = 0$ is $\sqrt{\frac{1}{2} (a + b)^{2} - 4 c}$ .
If the line $p x + q y + r = 0$ touches the circle $x^{2} + y^{2} = a^{2}$ then prove that $r^{2} = a^{2} (p^{2} + q^{2})$ .
Prove that the line $4 x - 3 y + 23 = 0$ touches the circle $x^{2} + y^{2} - 6 x + 10 y = 66$ .
Show that for all values of $θ$ , $x \sin θ - y \cos θ = a$ touches the circle $x^{2} + y^{2} = a^{2}$ .
If $l x + m y = 1$ touches the circle $x^{2} + y^{2} = a^{2}$ , prove that the point $(l, m)$ lies on the circle $x^{2} + y^{2} = a^{- 2}$ .
Find the value of $λ$ so that the line $3 x - 4 y = λ$ may touch the circle $x^{2} + y^{2} - 4 x - 8 y - 5 = 0$ .
Show that the line $(x - 1) \cos θ + (y - 1) \sin θ = 1$ touches a circle for all values of $θ$ . Find the circle.
Find those tangents to the circle $x^{2} + y^{2} = 16$ which are parallel to $3 x - 16 y = 10$ .
Find the equation of the tangents to the circle $x^{2} + y^{2} - 2 x - 4 y - 4 = 0$ which are (i) parallel, (ii) perpendicular to the line $3 x - 4 y - 1 = 0$ .
Show that the line $7 y - x = 5$ touches the circle $x^{2} + y^{2} - 5 x + 5 y = 0$ and find the equation of the other parallel tangent.
Find the equation of the tangent lines to the circle $x^{2} + y^{2} = 15$ which are perpendicular to the line $4 x - y + 6 = 0$ .
Find the equation of tangents to the circle $x^{2} + y^{2} - 6 x + 4 y - 3 = 0$ , which are perpendicular to the line $y = 2 x - 1$ .
Find the equation of the tangents to the circle $x^{2} + y^{2} = 25$ , which make an angle of $60^{\circ}$ with the positive direction of $x$ -axis.
Find the equation of the family of circles which touch the pair of straight lines $x^{2} - y^{2} + 2 y - 1 = 0$ .
Examine if the two circles $x^{2} + y^{2} - 2 x - 4 y = 0$ and $x^{2} + y^{2} - 8 y - 4 = 0$ touch each other externally or internally.
Prove that the circle $x^{2} + y^{2} + 2 a x + c^{2} = 0$ and $x^{2} + y^{2} + 2 b y + c^{2} = 0$ touch each other if $\frac{1}{a^{2}} + \frac{1}{b^{2}} = \frac{1}{c^{2}}$ .
Prove that $x^{2} + y^{2} = a^{2}$ and $(x - 2 a)^{2} + y^{2} = a^{2}$ are two equal circles touching each other. Find the equation of circles of equal radius touching both the circles.
Prove that the circles $x^{2} + y^{2} + 2 x + 2 y + 1 = 0$ and $x^{2} + y^{2} - 4 x - 6 y - 3 = 0$ touch each other.
Prove that the condition for the circles $(x - a)^{2} + (y - b)^{2} = c^{2}$ and $(x - b)^{2} + (y - a)^{2} = c^{2}$ to touch each other is $a = b \pm \sqrt{2} 𝑐$ .
Prove that the circles $x^{2} + y^{2} + 2 u x + 2 v y = 0$ and $x^{2} + y^{2} + 2 u_{1} x + 2 v_{1} y = 0$ touch each other if $u v_{1} = u_{1} v$ .
Find the length of the chord of the circle $x^{2} + y^{2} = 4$ through $(1, \frac{1}{2})$ , which is of minimum length.
Find the angle that the chord of circle $x^{2} + y^{2} - 4 y = 0$ along the line $x + y = 1$ subtends at the circumference the larger segment.
Prove, analytically, that the angle in a semi-circle is a right angle.
A circle of diameter $13$ m with the center $O$ coinciding with the origin of coordinate axes, has the diameter $A B$ on the 𝑥-axis. If the length of the chord $A C$ be $5$ m, find the equation of pair of lines $B C$ , $C$ having two possible positions.
Show that the least chord of the circle $x^{2} 2 + y^{2} + 2 g x + 2 f y + c = 0$ which passes through the internal point $(α, β)$ is equal to
$2 \sqrt{- (α^{2} + β^{2} + 2 g α + 2 f β + c)}$ .
If the line $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ cut the coordinate axes in concyclic points prove that $a_{1} a_{2} = b_{1} b_{2}$ .
The chord along the line $y - x = 3$ of the circle $x^{2} + y^{2} = k^{2}$ , subtends an angle of $30^{\circ}$ in the major segment of the circle cut off by the chord. Find $k$ .
Prove that the tangent to the circle $x^{2} + y^{2} = 5$ at the point $(1, - 2)$ also touches the circle $x^{2} + y^{2} - 8 x + 6 y + 20 = 0$ and find its point of contact.
rove that the equation $x^{2} + y^{2} - 2 x - 2 λ y - 8 = 0$ , where $λ$ is a parameter, represents a family of circles passing through two fixed points $A$ and $B$ on the $x$ -axis. Also, find the equation of that circle of the family the tangents to which at $A$ and $B$ meet on the line $x + 2 y + 5 = 0$ .
$A B$ is a diameter of a circle. $C D$ is a chord parallel to $A B$ and $2 C D = A B$ . The tangent at $B$ meets the line $A C$ produced at $E$ . Prove that $A E = 2 A B$ .
Two parallel tangents to a given circle are cut by a third tangent at point $A$ and $B$ . If $C$ be the center of the given circle, prove that $∠ A C B$ is a right angle.
A circle of radius $5$ meters is having its center at $A$ at the origin. Two circles II and III with centers $B$ and $C$ of radii $3$ and $4$ meters respectively touch the circle I and also touch the $x$ -axis to the right of $A$ . Find the equation to any two common tangents to the circles II and III.
Find the equation of the normal to the circle $x^{2} + y^{2} - 2 x - 4 y + 3 = 0$ at the point $(2, 3)$ .
The extremeties of a diagonal of a rectangle are $(- 4, 4)$ and $(6, - 1)$ . A circle circumscribes the rectangle and cuts an intercept $A B$ on the $y$ -axis. Find the area of the triangle formed by $A B$ and the tangents to the circle at $A$ and $B$ .
Find the equation of the tangent to the circle $x^{2} + y^{2} - 4 x - 6 y = 12$ at the point $(- 1, - 1)$ .
Prove that the tangents to the circle $x^{2} + y^{2} - 7 x - 5 y + 18 = 0$ at the point $(4, 3)$ and $(3, 2)$ are parallel to each other.
Find the equations of the tangents to the circle $x^{2} + y^{2} = 169$ at $(5, 12)$ and $(12, - 5)$ and prove that they cut at right angles. Find their point of intersection also.
The tangent at the point $(α, β)$ to the circle $x^{2} + y^{2} = r^{2}$ cut the axes of coordinates in $A$ and $B$ . Prove that the area of the $△ O A M$ is $\frac{1}{2} . \frac{r^{4}}{| α β |}$ , $O$ being the origin.
Let 𝐴 be the center of the circle $x^{2} + y^{2} - 2 x - 4 y - 20 = 0$ . Suppose that the tangents at the point $B (1, 7)$ and $D (4, - 2)$ on the circle meet at the point $C$ . Find the area of the quadrilateral $A B C D$ .
Prove that the line $x + y = 5$ touches the circle $x^{2} + y^{2} - 2 x - 4 y + 3 = 0$ . Find the point of contact.
Prove that the tangent to the circle $x^{2} + y^{2} = 5$ at the point $(1, - 2)$ also touches the circle $x^{2} 2 + y^{2} - 8 x + 6 y + 20 = 0$ , and find its point of contact.
Show that the circles $x^{2} 2 + y^{2} - 10 x + 4 y - 20 = 0$ and $x^{2} + y^{2} + 14 x - 6 y + 22 = 0$ touch each other. Find the coordinates of the point of contact and the equation of the common tangent at the point of contact.
Prove that the line $y = x + 2$ touches the circle $x^{2} + y^{2} = 2$ . Find the point of contact.
Find the condition that the straight line $l x + m y + n = 0$ should touch the circle $x^{2} 2 + y^{2} + 2 g x + 2 f y + c = 0$ and also find the coordinates of the point of contact.
If the line $3 x + 4 y = k$ touches the circle $x^{2} + y^{2} = 10 x$ , find the value of $k$ and also the point of contact.
Find the equation of the normal to the circle $x^{2} + y^{2} = 5$ at the point $(1, 2)$ .
Find the equation of the normal to the circle $x^{2} + y^{2} = 2 x$ , which is parallel to the line $x + 2 y = 3$ .
The point $(1, 4)$ is inside the circle $S$ whose equation is of the form $x^{2} + y^{2} - 6 x - 10 y + k = 0$ . What are the possible values of $k$ if the circle $S$ neither touches the axes not cuts the them?
Find the length of the tangent drawn from the point $(5, 1)$ to the circle $x^{2} + y^{2} + 6 x - 4 y - 3 = 0$ .
Distances from the origin to the centers of the three circles $x^{2} 2 + y^{2} 2 - 2 λ x = c^{2}$ , where $c$ is a constant and $λ$ a vairable are in G.P. Prove that the lengths of tangents drawn from any point on the circle $x^{2} + y^{2} = c^{2}$ to the three circles are also in G.P.
From a point $P$ tangents drawn to the circles $x^{2} + y^{2} + x - 3 = 0, 3 x^{2} + 3 y^{2} - 5 x + 3 y = 0$ and $4 x^{2} + 4 y^{2} + 8 x + 7 y + 9 = 0$ are of equal lengths. Find the equation of the circle thorugh $P$ which touches the line $x + y = 5$ at the point $(6, - 1)$ .
If the length of the tangent from $(f, g)$ to the circle $x^{2} + y^{2} = 6$ be twice the length of the tangent from $(f, g)$ to the circle $x^{2} + y^{2} + 3 x + 3 y = 0$ then will $f^{2} + g^{2} + 4 f + 4 g + 2 = 0$ ?
If the length of the tangent from a point $(f, g)$ to the circle $x^{2} + y^{2} = 4$ be four times the length of the tangent from it to the circle $x^{2} + y^{2} = 4 x$ , show that $15 f^{2} + 15 g^{2} - 64 f + 4 = 0$ .
Show that the length of the tangent from any point on the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ to the circle $x^{2} + y^{2} + 2 g x + 2 f y + c_{1} = 0$ is $\sqrt{c_{1} - 1}$ .
Find the point from which the tangents to the three circles $x^{2} + y^{2} = 1, x^{2} + y^{2} - 8 x + 15 = 0$ and $x^{2} 2 + y^{2} + 10 y + 24 = 0$ are equal in length.
Find the point from which the tangents to the three circles $x^{2} + y^{2} - 4 x + 7 = 0, 2 x^{2} + 2 y^{2} - 3 x + 5 y + 9 = 0$ and $x^{2} + y^{2} + y = 0$ are equal in length. Find the length as well.
If $A_{i}$ is the center of the circle $x^{2} + y^{2} + 2 g_{i} x + 5 = 0$ and 𝑡𝑖 is the length of the tangents from any point to this circle $i = 1, 2, 3$ ; then show that $(g_{2} - g_{3}) t_{1}^{2} + (g_{3} - g_{1}) t_{2}^{2} + (g_{1} - g_{2}) 𝑡_{3}^{2} = 0$ .
Show that if the length of the tangent from a point $P$ to the circle $x^{2} 2 + y^{2} = a^{2}$ be four times the length of the tangent from it to the circle $(x - a) 2 + y^{2} = a^{2}$ , then $P$ lies on the circle $15 x^{2} + 15 y^{2} - 32 a x + a^{2} = 0$ .
Find the equation of the pair of tangents drawn to the circle $x^{2} + y^{2} - 2 x + 4 y = 0$ from point $(0, 1)$ .
If from any point on the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ , tangents are drawn to the circle $x^{2} + y^{2} + 2 g x + 2 f y + c \sin^{2} α + (f^{2} + g^{2}) c o s^{2} α = 0$ , show that the angle between the tangents is equal to $2 α$ .
Find the equations of tangents to the circle $x^{2} + y^{2} = 25$ which pass through $(1, - 7)$ and show that they are at right angles.
Show that two tangents can be drawn from the point $(9, 0)$ to the circle $x^{2} + y^{2} = 16$ ; also find the equation of the pair of tangents and the angle between them.
Find the equations of the tangents through $(7, 1)$ to the circle $x^{2} + y^{2} = 25$ .
Find the equation of the pair of tangents from the origin to the circle $x^{2} + y^{2} + 2 g x + 2 f y + k^{2} = 0$ , and show that their intercept on the line $y = h$ is $\frac{2 h k}{k^{2} - g^{2}}$ times the radius of the circle.
Find the equation of the chord of the circle $x^{2} + y^{2} + 6 x + 8 y - 11 = 0$ , whose middle point is $(1, - 1)$ .
Find the equation of the chord of the circle $x^{2} + y^{2} + 6 x + 8 y + 9 = 0$ whose mid-point is $(- 2, - 3)$ .
Find the coordinates of the mid-point of the chord which the circle $x^{2} + y^{2} + 4 x - 2 y - 3 = 0$ cuts off on the line $y = x + 2$ .
Find the equation of the chord fo contact of the tangents drawn from $(1, 2)$ to the circle $x^{2} + y^{2} - 2 x + 4 y + 7 = 0$ .
Tangents are drawn from the point $(h, k)$ to the circle $x^{2} + y^{2} = a^{2}$ . Prove that the area of the triangle formed by them and their chord of contact is $\frac{a {(h^{2} + k^{2} - a^{2})}^{\frac{3}{2}}}{h^{2} + k^{2}}$ .
The chord of contact of tangents from a point on the circle $x^{2} + y^{2} = a^{2}$ to the circle $x^{2} + y^{2} = b^{2}$ touches the circle $x^{2} + y^{2} = c^{2}$ . Show that $a, b, c$ are in G.P.
Tangents are drawn to the circle $x^{2} + y^{2} = 12$ at the points where it is met by the circle $x^{2} + y^{2} - 5 x + 3 y - 2 = 0$ ; find the points of intersection of these tangents.
Find the equation of the chord of contact of the tangents drawn from an external point $(- 3, 2)$ to the circle $x^{2} + y^{2} + 2 x - 3 = 0$ .
Find the equation of the chord of contact of the tangents drawn from $(5, 3)$ to the circle $x^{2} + y^{2} = 25$ .
Find the coordinates of the point of intersection of tangents at the points where the line $2 x + y + 12 = 0$ meets the circle $x^{2} + y^{2} - 4 x + 3 y - 1 = 0$ .
The lnegth of tangents from the two given points to a given points to a given circle are $t_{1}$ and $t_{2}$ . If the tow given points are conjugate to each other w.r.t. the given circle, prove that the distance between the points will be $\sqrt{t_{1}^{2} + t_{2}^{2}}$ .
Find the area of the triangle formed by the tangents drawn from the point $(4, 6)$ to the circle $x^{2} + y^{2} = 25$ and their chord of contact.
If $O P$ and $O Q$ are the tangents from the origin to the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ , where $P$ and $Q$ are the points of contact, show that the equation of the circumcircle of the $△ O P Q$ is $x^{2} + y^{2} + g x + f y = 0$ .
Find the point of contact of the tangents to the circle $x^{2} + y^{2} = 25$ that pass through the point $(7, 1)$ and give the equation of tangents.
Find the equation of the polar of the point $(2, - 1)$ w.r.t. the circle $x^{2} + y^{2} - 3 x + 4 y - 8 = 0$ .
Find the pole of the line $3 x + 5 y + 17 = 0$ w.r.t. the circle $x^{2} + y^{2} + 4 x + 6 y + 9 = 0$ .
Show that the polars of the point $(1, - 2)$ w.r.t. the circle $x^{2} + y^{2} + 6 y + 5 = 0$ and $x^{2} + y^{2} + 2 x + 8 y + 5 = 0$ coincide. Prove that there is another point, the polars of which w.r.t. these circles are the same and find its coordinates.
Let $C$ be the center of a circle. The lines $L_{1}$ and $L_{2}$ are the polars of the points $A$ and $B$ respectively. w.r.t. the circle. Perpendiculars $A M$ and $B N$ are dropped from $A$ to the line $L_{2}$ and from $B$ to $L 1$ . Prove that $C A : C B = A M : B N$ .
Find the polar of the point of intersection of the line $4 x - y = 11$ and $x - 2 y = 1$ w.r.t. the circle $x^{2} + y^{2} = 7$ .
Find the polar of the point $(4, - 1)$ w.r.t. the circle $2 x^{2} + 2 y^{2} = 11$ .
Find the polar of the point $(1, - 5)$ w.r.t. the circle $x^{2} + y^{2} - 8 x + 6 y + 4 = 0$ .
Prove that the polar of the point $(p, q)$ w.r.t. the circle $x^{2} + y^{2} = a^{2}$ touches $(x - c)^{2} + (y - d)^{2} = b^{2}$ if $b^{2} (p^{2} + q^{2}) = {(a^{2} - c p - d q)}^{2}$ .
Show that the polar of the origin w.r.t. the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ touches the circle $x^{2} + y^{2} = a^{2}$ if $𝑐 2 = a^{2} (f^{2} + g^{2})$ .
Prove that if the pole of a straight line w.r.t. the circle $x^{2} + y^{2} = c^{2}$ lies on the circle $x^{2} + y^{2} = 9 c^{2}$ , the line is a tangent to the circle $9 x^{2} + 9 y^{2} = c^{2}$ .
Find the pole of the straight line $9 x + y - 28 = 0$ w.r.t. the circle $2 x^{2} + 2 y^{2} - 3 x + 5 y - 7 = 0$ .
Find the pole of the straight line $2 x - y + 10 = 0$ w.r.t. the circle $x^{2} + y^{2} - 7 x + 5 y - 1 = 0$ .
Find the pole of the line $a x + b y + c = 0$ w.r.t. the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ .
Prove that the polar of any point w.r.t. a circle is perpendicular to the line joining the point and center of the circle.
Prove that the polar of a given point w.r.t. any of the circles $x^{2} + y^{2} + 2 p x + c = 0$ , where $p$ is a variable, always passes through a fixed point.
If the polar of the point $(α, β)$ w.r.t. the circle $x^{2} + y^{2} = a^{2}$ touches the circle $(x - a)^{2} + y^{2} = a^{2}$ , show that $(α, β)$ is on the curve given by $y^{2} + 2 a x = a^{2}$ .
Verify that the three points $(x_{1}, y_{1}), (x_{2}, y_{2})$ and $(x_{3}, y_{3})$ will be collinear if and only if their polars w.r.t. the circle $x^{2} + y^{2} = a^{2}$ are concurrent.
Show that the circles $x^{2} + y^{2} - 2 x - 6 y - 12 = 0$ and $x^{2} + y^{2} + 6 x + 4 y - 6 = 0$ cut each other orthogonally.
If $S = 0$ and $S_{1} = 0$ are the two circles with radii $a$ and $a_{1}$ respectively. Show that the circles $\frac{S}{a} \pm \frac{S_{1}}{a_{1}} = 0$ intersect at right angles.
Prove that the two circles pass through the points $(0, 0)$ and $(0, - a)$ and touch the line $y = m x + c$ will cut orthogonally if $c^{2} = a^{2} (2 + m^{2})$ .
Obtain the equation of the circle orthogonal to both the circles $x^{2} + y^{2} + 3 c - 5 y + 6 = 0$ and $4 x^{2} + 4 y^{2} - 28 x + 29 = 0$ , and whose center lies on the line $3 x + 4 y + 1 = 0$ .
Prove that a circle cutting the circle $x^{2} + y^{2} = 4$ orthogonally and having its center on the line $2 x - 2 y + 9 = 0$ , passes through two fixed points, and find the points.
Prove that the general equation of circles cutting the circles $x^{2} + y^{2} + 2 g_{r} x + 2 f_{r} y + c_{r} = 0; r = 1, 2$ orthogonally is $| \begin{matrix} x^{2} + y^{2} & - x & - y \\ c_{1} & g_{1} & f_{1} \\ c_{1} & g_{1} & f_{1} \end{matrix} | + k | \begin{matrix} - x & - y & 1 \\ g_{1} & f_{1} & 1 \\ g_{2} & f_{2} & 1 \end{matrix} | = 0$ .
For what value of $k$ the circles $x^{2} + y^{2} + 5 x + 3 y + 7 = 0$ and $x^{2} + y^{2} - 8 x + 6 y + k = 0$ cut orthogonally.
Find the equation of the circle passing through the origin and cutting the circles $x^{2} + y^{2} - 4 x + 6 y + 10 = 0$ and $x^{2} + y^{2} + 12 y + 6 = 0$ at right angles.
Find the equation of the circle passing through the origin and has its center on the line $x + y + 4 = 0$ and cuts the circle $x^{2} + y^{2} - 4 x + 2 y + 4 = 0$ orthogonally.
If two circles cut a third circle orthogonally then prove that their common chord will pass through the center of the third circle.
If a circle cuts orthogonally three circles $S_{1} = 0, S_{2} = 0, S_{3} = 0$ ; prove that it cuts orthogonally any circle of the form $k S_{1} + l S_{2} + m S_{3} = 0$ .
Prove that the two circles each of which passes through the points $(0, k)$ and $(0, - k)$ and touches the line $y = m x + b$ will cut orthogonally if $b^{2} = k^{2} (2 + m^{2})$ .
Find the general equation of a circle cutting $x^{2} + y^{2} = c^{2}$ orthogonally and show that if it passes through the point $(a, b)$ will also pass through the point $(\frac{c^{2} a}{a^{2} + b^{2}}, \frac{c^{2} b}{a^{2} + b^{2}})$ .
If $P$ and $Q$ be a pair of conjugate points w.r.t. a circle $S$ , prove that the circle on $P Q$ as a diameter cuts the circle $S$ orthogonally.
Prove that the circle $x^{2} + y^{2} - 6 x - 4 y + 9 = 0$ bisects the circumference of the circle $x^{2} + y^{2} - 8 x - 6 y + 23 = 0$ .
Find the equation of a circle which is coaxal with the circles $2 x^{2} + 2 y^{2} - 2 x + 6 y - 3 = 0$ and $x^{2} + y^{2} + 4 x + 2 y + 1 = 0$ . It is given that the center of the circle to be determined lies on the radical axis of these circles.
If the radical axis of the circles $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ and $2 x^{2} + 2 y^{2} + 3 x + 8 y + 2 c = 0$ touches the circle $x^{2} + y^{2} + 2 x - 2 y + 1 = 0$ , show that either $g = \frac{3}{4}$ or $f = 2$ .
Find the general equation of circles, any two of which have the same radical axis as that of the circles $x^{2} + y^{2} + 2 x + 4 y - 6 = 0$ and $x^{2} + y^{2} = 4$ .
The equations of three circles are $x^{2} + y^{2} = 1, x^{2} + y^{2} - 8 x + 15 = 0, x^{2} + y^{2} + 10 y + 24 = 0$ . Determine the coordinates of the point such that the tangents drawn from it to the three circles are equal in length.
The polars of a point $P$ w.r.t. two given circles meet in a point $Q$ ; show that the radical axis of the circles bisedct the line $P Q$ .
Show that the locus of a point such that the ratio of its distances from two given points is a constant, is a circle. Hence, show that this circle cannot pass through the given points.
Two rods of lengths $a$ and $b$ slide along the axes in a manner that their ends are always concyclic. Find the locus of the center of the circle passing through these ends.
Two straight lines rotate about two fixed points. If they start from their positions of coincidence such that one rotates at the rate double that of the other. Prove that locus of their point of intersection is a circle.
A circle of radius $r$ passes through the origin $O$ , and cuts the axes at $A$ and $B$ . Let $P$ be the foot of the perpendicular from the origin to the line $A B$ . Find the equation of the locus of $P$ .
Show that the locus of points from which the tangents drawn to a circle are orthogonal, is a concentric circle or find the equation of the cirector circle of the circle $x^{2} + y^{2} = a^{2}$ .
Find the locus of the point of the point of intersection of tangents to the circle $x = a \cos θ, y = a \sin θ$ at points whose parametric angles differ by $\frac{π}{3}$ .
The circles $x^{2} + y^{2} + 2 a x - c^{2} = 0$ and $x^{2} + y^{2} + 2 b x - c^{2} = 0$ intersect at $A$ and $B$ . A line through $A$ meets one circle at $P$ and a parallel line through $B$ meets at the other circle at $Q$ . Show that the locus of the mid-point of $P Q$ is a circle.
Find the condition that the chord of contact of tangents from the point $(α, β)$ to the circle $x^{2} + y^{2} = a^{2}$ should subtend a right angle at the center. Hence, find the locus of $(α, β)$ .
A tangent is drawn to each of the circle $x^{2} + y^{2} = a^{2}$ and $x^{2} + y^{2} = b^{2}$ . Show that if the two tangents are mutually perpendicular, the locus of their point of intersection is a cricle concentric with the given circle.
S how that the locus of the point, the tangents from which to the circle $x^{2} + y^{2} = a^{2}$ include a constant angle $α$ is ${(x^{2} + y^{2} - 2 a^{2})}^{2} \tan^{2} α = 4 a^{2} (x^{2} + y^{2} - a^{2})$ .
A straight line passes through the fixed point $(h, k)$ . Find the locus of the foot of the perpendicular drawn to it from the origin.
$O$ is a fixed point and $A P$ and $B Q$ are two fixed parallel straight lines; $B O A$ is perpendicular to both and $∠ P O Q$ is a right angle. Prove that the locus of the foot of the perpendicular drawn from $O$ upon $P Q$ is the circle on $A B$ as diameter.
A variable circle passes through the point $P (1, 2)$ and touches the $x$ -axis; show that the locus of the other end of the diameter through $P$ is $(x - 1)^{2} = 8 y$ .
Find the locus of a point, which is such that the lengths of the tangents from it to two concentric circles $x^{2} + y^{2} = a^{2}$ and $x^{2} + y^{2} = b^{2}$ vary inversely as their radii.
A point moves such that the sum of squares of its distances from the sides of a square of side unity is equal to $9$ . Show that the locus of a circle whose center is coincides with the center of the square. Also, find its radius.
Find the locus of the center of the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ when the pair of tangents drawn from the origin to the circle are perpendicular to each other.
Determine the locus of centers of the circles which touches the two circles $x^{2} + y^{2} = a^{2}$ and $x^{2} + y^{2} = 4 a x$ externally.
Find the locus of the centers of the circles which cut the circles $x^{2} + y^{2} + 4 x - 6 y + 9 = 0$ and $x^{2} + y^{2} - 4 x + 6 y + 4 = 0$ orthogonally is $8 x - 12 y + 5 = 0$ .
Find the locus of the foot of the perpendicular drawn from a fixed point on the $x$ -axis to any tangent to the circle $x^{2} + y^{2} = a^{2}$ .
The tangent at any point $P$ on the circle $x^{2} + y^{2} = 2$ cuts the axes in $L$ and $M$ . Find the locus of the middle point of $L M$ .
A triangle has two of its sides along the axes, its third side touches the circle $x^{2} + y^{2} - 2 a x - 2 a y + a^{2} = 0$ . Find the equation of the locus of the circumcenter of the triangle.
The point $A (1, 5)$ is joined to any point $P$ of the circle $x^{2} + y^{2} = 4$ . Find the locus of the middle point of $A P$ as $P$ moves on the circle.
Chords of the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ drawn through a fixed point $A (a, b)$ . Find the locus of the mid-points of these chords and interpret the locus.
A straight line moves so that the algebraic sum of the perpendiculars drawn to it from two fixed points is constant, show that the line always touches a fixed circle.

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