Question

Assertion :Statement-$1$: Equation of the polar of $P(-7,-9)$ with respect to the circle $x^2+y^2-12x-8y-48=0$ is $13x+l3y-30=0$ Reason: Statement-2: $P$ is a point outside the circle $C$. A line passing through $P$ meets the circle $C$ at $Q$ and $R$. The tangents at the extremities of the Chord $QR$ of the circle $C$ intersect at $S$. The locus of $S$, for different Chords of the circle $C$ passing through $P$, is called the polar of $P$ with respect to the circle $C$ and its equation is same as the chord of contact of $P$ with respect to the circle $C$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Statement-2 is True.

Let $QR$ be a chord of a circle passing through the point $P(x_1,y_1),$ and let the tangents at the extremities $Q$ and $R$ of this chord intersect at the point $L(h,k)$

Then locus of $L$ is called the polar of $P$ with respect to the circle, and $P$ is called the pole of its polar.

(1) Equation of the polar of $P(x_1,y_1),$ with respect to the cirlce $S\equiv x^2+y^2+2gx+3fy+c=0,$ is $T=0,$ where $T$ is as defined in (8) above.

(2) If the polar of $P$ with respect to a circle passes through $S,$ then the polar of $S$ with respect to the same circle passes through $P.$

Two such points $P$ and $S$ are called conjugate points of the circle.

Using it in statements-1, the required equation is

$-7x-9y-6(x-7)-4(y-9)-48=0$ or $13x+13y-30=0$

showing that statement -1 is also true.