Question

Prove that the polars of the point (1, - 2) with respect to the circles whose equations are
$x^2+y^2+6y+5=0$ and $x^2+y^2+2x+8y+5=0$
coincide ; prove also that there is another point where the polars of which with respect to these circles are the same and find its coordinates.

Answer 1

For the general equation of circle

The polar of circle w.r.t the pole $(x_1,y_1)$ is given by

$x{x}_1+y{y}_1+g(x+x_1)+f(y+y_1)+c=0$

For the first circle, equation of polar

$x-2y+0+3(-2+y)+5=0$

$x+y=1$ .....(1)

For second circle

$x-2y+(x+1)+4(y-2)+5=0$

$x+y=1$ .......(2)

Equation (1) and (2) are same so the polars coincide

Hence proved

Let the other point w.r.t which the polars are same be $(x_2,y_2)$

For first circle, equation of polar

$x{x}_2+y{y}_2+3(y+y_2)+5=0$

$x{x}_2+y(y_2+3)+3y_2+5=0$.......(3)

For second circle

$x{x}_2+y{y}_2+(x+x_2)+4(y+y_2)+5=0$

$x(x_2+1)+y(y_2+4)+x_2+4y_2+5-0$.........(4)

comparing equation (3) and (4)

$(x_2,y_2)=(2,-1)$