Question

True or False

Statement: Polar of the point $P(x_1,y_1)$ with respect to the circle $x^2+y^2=a^2$ is $x{x}_1+y{y}_1=a^2$ (P is inside the circle)

• A. True
• B. False

Answer 1

The correct option is A

True

The given situation can be represented as below. Let a random straight line passing through P intersects the circle at R and S. The tangents at R and S intersects at T. Polar is the locus of all such $T_s^{\prime }.$

Let $T\equiv (h,k)$

Then RS is chord of contact of T which is given by equation,

$hx+ky=a^2$ This passes through $(x_1,y_1)$

$\Rightarrow x_{1}h+y_{1}h=a^2$

Hence the locus of T is,

$x{x}_1+y{y}_1=a^2,$ which implies the given statement to be true.