# Circles in Conic Sections

## Circle

• A circle is the set of all points in a plane that are equidistant from a fixed point in the plane.
• The fixed point is called the centre of the circle
• The distance from the centre to a point on the circle is called the radius of the circle
• The general equation of a circle with center $(h, k)$ and radius $r$ is:

$(x - h)^2 + (y - k)^2 = r^2$
• The special case where the center is the origin

$x^2 + y^2 = r^2$.

## General Equation of Circle

The General equation of Circle is
$x^2 + y^2 + 2gx + 2fy + c=0$
Center is given by (-g, -h)
$r=\sqrt {g^2 + f^2 -c}$

## Equation of Circle when points of extremities of diameter are Given

Here we have to find Equation of Circle with points $P(x_1, y_1)$ and $Q(x_2, y_2)$ as extremities of diameter
Let A(h,k) be any point on the circle
Equation of line AP will be
$y -y_1= \frac {k-y_1}{h-x_1} (x- x_1)$
Equation of line AQ will be
$y -y_2= \frac {k-y_2}{h-x_2} (x- x_2)$
Since two lines are Perpendicular in semi circle
$\frac {k-y_1}{h-x_1}. \frac {k-y_2}{h-x_2}=-1$ $(h-x_1)(h-x_2)+ (k-y_1)(k-y_2)=0$
So for any point x, y we can write as
$(x-x_1)(x-x_2)+ (y-y_1)(y-y_2)=0$

## Position of a Point with Respect to a Circle

Let the circle Equation $S ≡ x^2+y^2+2gx+2fy+c = 0$
A point $(x_1,y_1)$ lies outside the circle if $S_1 ≡ x_1^2+y_1^2+2gx_1+2fy_1+c$ is positive ($S_1 > 0$).
A point $(x_1,y_1)$ lies inside the circle if $S_1 ≡ x_1^2+y_1^2+2gx_1+2fy_1+c$ is negative ($S_1 < 0$).
A point $(x_1,y_1)$ lies one the circle if $S_1 ≡ x_1^2+y_1^2+2gx_1+2fy_1+c$ is Zero ($S_1 = 0$).

## Line and a Circle

Let L = 0 be a line, and S = 0 be a circle; if r is the radius of a circle and p is the length of the perpendicular from the centre of the circle to the line, then if
p=r, => Line touches the circle.
p>r, => Line is outside the circle.
p Line is the chord of the circle.
p=0, => Line is the diameter of the circle.

## Equation of Tangent to the circle

Equation of tangent at $P(x_1,y_1)$ in a circle
Case A
For Circle as center as Origin
$x^2 + y^2 =a^2$
Equation of tangent
$xx_1 + yy_1=a^2$

Case B
For general Circle
$x^2 + y^2 + 2gx + 2fy + c=0$
Equation of tangent is
$xx_1 + yy_2 + g(x+x_1) + f(y+y_1) + c=0$

## Condition for Tangency

A line L = 0 touches the circle S = 0, if the length of the perpendicular drawn from the centre of the circle to the line is equal to the radius of the circle, i.e., p = r. This is the condition of tangency for the line L = 0.
The circle $x^2+y^2 = a^2$ touches the line $y = mx+c$ if $c = \pm a \sqrt {(1+m^2)}$
(a) If $a^2(1+m^2) – c^2 > 0$, the line will meet the circle at real and different points.
(b) If $c^2 = a^2(1+m^2)$, the line will touch the circle.
(c) If $a^2(1+m^2) – c^2 <0$, the line will meet the circle at two imaginary points.
From this condition of tangency
Line $y = mx \pm a \sqrt {(1+m^2)}$ is a tangent of the circle $x^2+y^2 =a^2$

## Equation of Normal to Circle

Normal to any point on the circle is the line which passes through point and its perpendicular to the tangent at that point
Case A
For Circle as center as Origin
$x^2 + y^2 =a^2$
Equation of Normal
$xy_1 - yx_1=0$

Case B
For general Circle
$x^2 + y^2 + 2gx + 2fy + c=0$
Equation of Normal is
$y-y_1 = \frac {y_1+f}{x1+g}(x-x_1)$

## Length of Tangent of Circle

Two tangents can be drawn from a point $P(x_1, y_1)$ to a circle. Let PQ and PR be two tangents drawn from P(x1, y1) to the circle x2+y2+2gx+2fy+c = 0. Then, PQ = PR is called the length of the tangent drawn from the point and is given by
$PQ = PR = \sqrt {(x_1^2+y_1^2+2gx_1+2fy_1+c)}$
For Circle as center as Origin
$x^2 + y^2 =a^2$ $PQ = PR = \sqrt {(x_1^2+y_1^2-a^2)}$

## Pair of Tangents to Circle

Two tangents, PQ and PR, can be drawn from a point $P(x_1, y_1)$ to the circle $S = x^2+y^2+2gx+2fy+c = 0$.
The combined equation is given by
$(x^2+y^2+2gx+2fy+c)(x_1^2+y_1^2+2gx_1+2fy_1+c)= (xx_1 + yy_2 + g(x+x_1) + f(y+y_1) + c)^2$
This can be written like below to remember easily
$SS_1 = T^2$
Where S = 0 is the equation of the circle, T = 0 is the equation of tangent at $(x_1, y_1)$, and S1 is obtained by replacing x by $x_1$ and y by $y_1$ in S.

## Chord of Contact

The chord joining the two points of contact of tangents to a circle drawn from any point P is called the chord of contact of P with respect to the given circle.
Case A
For Circle as center as Origin
$x^2 + y^2 =a^2$
Equation of Chord of Contact
$xx_1 + yy_1=a^2$
Case B
For general Circle
$x^2 + y^2 + 2gx + 2fy + c=0$
Equation of Chord of Contact
$xx_1 + yy_2 + g(x+x_1) + f(y+y_1) + c=0$
If the point (x1, y1) lies on the circle, then the chord of contact coincides with the equation of the tangent.

## Equation of chord whose midpoint is given

The equation of the chord of the circle $x^2+y^2 = a^2$ whose midpoint $P(x_1, y_1)$ given is
$y-y_1 = \frac {-x_1}{y_1}(x-x_1)$ or $xx_1+yy_1 = x_1^2+y_1^2$, which can be represented by $T = S_1$

## Common chord of two circles

The line joining the points of intersection of two circles is called the common chord.
Let $S_1$ and $S_2$ be the 2 circles
$S_1 = x^2+y^2+2g_1x+2f_1y+c_1 = 0$ and $S_2= x^2+y^2+2g_2x+2f_2y+c_2 = 0$
Then, the equation of the common chord is $S_1-S_2 = 0$
$2x(g_1-g_2) + 2y(f_1-f_2) + c_1-c_2 = 0$

## Family of Circles

Family of circles passing through the point of intersection of a line and circle
The required equation for the family of circles passing through the point of intersection of circle S = 0 and line L = 0 is given by $S+ \lambda L = 0$, where $\lambda$ is a parameter.

Family of circles passing through the point of intersection of two circles
The equation of the family of circles passing through the point of intersection of two circles $S_1 = 0$ and $S_2 = 0$ is given by $S_1 + \lambda S_2 = 0$. Where $\lambda\ne -1$.
Note that both circles should be in the given format of $S_1$ and $S_2$.

## Orthogonal Circles

"Orthogonal circles" refers to two circles that intersect at a right angle. This means that the tangent lines to each circle at the points of intersection are perpendicular to each other.
If $r_1$ and $r_2$ are the radius of the circles and d is the distance between the center, then circles will be orthogonal if
$d^2 = r_1^2 + r_2^2$
This is because the radii of the circles at the points of intersection will form a right angle with the line connecting the centers.

For the General equation of Circles
$S_1 = x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0$
$S_2 = x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0$
$S_1$ and $S_2$ are orthogonal if $2g_1g_2 + 2f_1f_2 = c_1+ c_2$

## Contact of Circles

Case 1

If the sum of the radii of two circles is equal to the distance between the centres, then the circles touch externally. The circles will satisfy the condition
$c_1c_2 = r_1 + r_2$

Case 2

If the difference between the radii and the distance between the centres are equal, then the circles touch internally. The circles will satisfy the condition
$c_1c_2 = r_1 – r_2$