Circles

1. Definition: A circle is the locus of a point which lies in a plane in such a way that its distance from a fixed point in the plane is constant. The fixed point is called the radius of the circle.
2. Equation of a circle: Different forms.

(A) Centre (h,k), radius a, ${(x-h)}^2$+${(y-h)}^2$= $a^2$ Centre (0,0), radius a, $x^2$+ $y^2$= $a^2$

(B) Centre (h,k) and passes through the origin $x^2$+ $y^2$- 2hx - 2ky= 0

(C) Centre (h,k) and touches the x- axis ${(x-h)}^2$+${(y-k)}^2$= $k^2$
or $x^2$+ $y^2$- 2hx - 2ky +

$h^2$= 0

D) Centre (h,k) and touches the y axis ${(x-h)}^2$+${(y-k)}^2$= $h^2$
or $x^2$+ $y^2$- 2hx - 2ky +$k^2$= 0

E) General Equation of a circle: $x^2$+ $y^2$+2gx + 2fy + c= 0

Centre (-g, -f) and radius $\sqrt{g^2+f^2-c}$

F) Length of intercepts

Intercept on x- axis= 2$\sqrt{g^2-c}$ ; Intercept on y- axis= 2$\sqrt{f^2-c}$

G) Circle whose diameter is the line joining ($x_1$, $y_1$) and ($x_2$, $y_2$) is (x- $x_1$) (x- $x_2$) +

(y- $y_1$) (y- $y_2$)= 0

H) Circle through three given points

Let the equation be $x^2$+ $y^2$+2gx + 2fy + c= 0. The points satisfy this equation and we

will get three equations in g, f, c. Solving them, there valves can be obtained.

I) Let $C_1$, $C_2$ denote the centre of two circles and $r_1$, $r_2$ denote their radii

(a) If $C_1$$C_2$ $>$ $r_1$+ $r_2$, then the two circles do not intersect with each other and four common tangents can be drawn to two circles.

(b) If $C_1$, $C_2$ = $r_1$+ $r_2$, circles touch each other externally. Then 3 tangents can be drawn common to the circles.

(c) $C_1$ $C_2$ = ${|r}_1$- $r_2$|, circels touch each other internally, one common tanget can be drawn.

(d) If ${|r}_1$- $r_2$| $<$ $C_1$$C_2$ $<$ $r_1$+ $r_2$ circles do not touch but cut each other. Here two common chord of the circles $S_1$= 0 and $S_2$= 0is $S_1$- $S_2$= 0

The chord joining th points of intersection of two given circles is called their common chord. Also its lenght AB= 2 AM= 2$\sqrt{r_1^2-p_1^2}$= 2$\sqrt{r_2^2-p_2^2}$

Where $p_1$ and $p_2$are the perpendicular distance of the centres from the common chord.

5. If AB= 0 then the two circles touch each other and in these case common chord becomes common tangent.

6. The equation of a circle passing through the intersection of S= 0 and given line P= 0 is S + $\lambda$ P= 0.

7. The equation of a circle passing through the intersection of the circles $S_1$= 0 and $S_2$= 0 is $S_1$ + $\lambda$ $S_2$ = 0

8. The power of the point ($x_1$, $y_1$) with respect to the circle S $\equiv$ $x^2$+ $y^2$ + 2gx + 2fy +c= 0 is : $S_1$ $\equiv$ $x_1^2$ + $y_1^2$ + 2${\mathrm{gx}}_1$+ 2${\mathrm{fy}}_1$ + c.

The point ($x_1$, $y_1$) lies outside, on or inside the given circle according as $S_1$ $>$ 0 = 0

or $<$ 0.

9. Tangent of the circle,

S= $x^2$+ $y^2$+ 2gx + 2fy + c= 0 at any point ($x_1$, $y_1$) is $T_1$= ${\mathrm{xx}}_1$+ ${\mathrm{yy}}_1$+g(x+$x_1$) + f(y+

$y_1$) + c= 0.

If the equation of the circle is $x^2$ + $y^2$= $a^2$ , then the equation of the tangent at ($x_1$,$y_1$

is ${\mathrm{xx}}_1$+ ${\mathrm{yy}}_1$= $a^2$.

10. Condition for the line y= mx + c to be a tangent to the circle $x^2$+$y^2$= $a^2$ is c= a $\sqrt{1+m^2}$, the equation of the tangent in terms of m is y = mx $\pm$ a $\sqrt{1+m^2}$

11. The length of the tangent from a given point ($x_1$, $y_1$) to the circle S= $x^2$+ $y^2$+ 2gx + 2fy + c= 0 is: $\sqrt{S_1}$= $\sqrt{x_1^2+y_1^2+2{\mathrm{gx}}_1+2{\mathrm{fy}}_1+c}$

12. Orthogonal circles: Two circles are said to cut orthogonally if the angle between them is a right angle. The conditions of two circles S= $x^2$+ $y^2$+ 2gx + 2fy + c= 0 and $S^1$= $x^2$+ $y^2$+ 2$g^1$x + 2$f^1$y + $c^1$= 0 to cut orthogonally is: 2g$g^1$+2f$f^1$= c + $c^1$

13. Radical Axis

The radical axis of two circles is the locus of a point which moves such that the lengths of tangents drawn from it

to the two circles are equal. ie, $S_1$- $S_2$= 0

If the circles touch, the radical axis becomes common tangent and if they intersect it becomes the equation of the

common chord. The radical axis is perpendicular to the line joining the centres of the two circles.

Radical Centre

The radical centre of three circles is the point of concurrence of the radical axes of the circles taken in pairs. The

length of the tangents drawn from the radical centre to the three circles are equal.

If two given circles intersect each other then the radical axis is the same as the common tangent. The pair of

tangents from (0,0) to the circle $x^2$+$y^2$+2gx + 2fy + c= 0 are at right angles if $g^2$+$f^2$= 2c.