Question

If (h, k) is a point from which the tangents to the three circles $x^2+y^2-4x+7=0$, $2x^2+2y^2-3x+5y+9=0$ and $x^2+y^2+y=0$ are equal in length. Find the value of h + k ___ .

Answer 1

Given circles

$s_1,x^2+y^2-4x+7=0$ - - - - - - -(1)

$s_2,x^2+y^2-\frac{3x}{2}+\frac{5y}{2}+\frac{9}{2}=0$ - - - - - - -(2)

$s_3,x^2+y^2+y=0$ - - - - - - - (3)

(h,k) ia a point from which the tangents to the three circles are equal in length .So,(h,k) is the radical

centre of the three circles.

So,radical axes of circle (1) and (2) is

$S_1-S_2=0$

$\frac{-5}{2}x-\frac{5}{2}y+\frac{5}{2}=0$

$-x-y+1=0$

$x+y=1$ - - - - - - -(4)

Radical axes of circle (2) and (3) is

$S_2-S_3=0$

$\frac{-3}{2}x+\frac{3}{2}y+\frac{9}{2}=0$

$-x+y+3$ - - - - - - -(5)

$x-y=3$

Solving equation 4 and 5

We get,

$2x=4$

$x=2$

$y=-1$

Coordinates of (h,k) is (2,-1)

Radicalcenter is (2,-1)

$h=2,k=-1$

$h+k=2-1=1$