Question

Assertion :Statement-1: The line $x+9y-12=0$ is a chord of contact of a point $P$ with respect to the circle $2x^2+2y^2-3x+5y-7=0$. Reason: Statement-2: The line joining the points of contacts of the tangents drawn from a point $P$ outside a circle to the circle is the chord of contact of $P$ with respect to the circle.

• 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 incorrect but Reason is correct
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is C Assertion is incorrect but Reason is correct

Statement-$2$ is true by definition of chord of contact of a point with respect to a circle. Now
let $P(h,k)$ be a point whose chord of contact is the line in statement $1$,

Then $2xh+2yk-\frac{3}{2}(x+h)+\frac{5}{2}(y+k)-7=0$ is same as $x+9y-12=0$

$\Rightarrow \frac{4h-3}{1}=\frac{4k+5}{9}=\frac{3h-5k+14}{12}$

$\Rightarrow h=1,k=1$ so the point $P$ is $(1,1)$

But $S(1,1)=2+2-3+5-7<0$ where $S=2x^2+2y^2-3x+5y-7$.
Showing that $P$ lies inside the circle $S=0$ and the given line cannot be a chord of contact of the circle $S=0$ for any point $P$ and thus the statement-$1$ is false.