Question

Assertion :Consider $L_1:2x+3y+p-3=0L_2:2x+3y+p+3=0,$ where p is a real number, and $C:x^2+y^2+6x-10y+30=0$ Statement-1: If line $L_1$ is a chord of circle $C$, then $L_2$ is not always a diameter of circle $C.$ Reason: If line $L_1$ is a diameter of circle $C$, then $L_2$ not a chord of 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 C Assertion is correct but Reason is incorrect

$C:(x+3{)}^2+(y-5{)}^2=4$ centre of $C$ is $(-3,5)$ and radius is $2.$
If $L_1$ is a chord of $C$ then $\frac{-6+15+p-3}{\sqrt{4+9}}<2$
$\Rightarrow (p+6{)}^2<52$ $p(p+12)-16<0\left(1\right)$
$L_2$ is a diameter of $C$ if $-6+15+p+3=0$ or if $p+12=0\left(2\right)$
For $p+12=0$, $\left(1\right)$ is satisfied.
So $L_2$ is a diameter of $C$ only when $p=-12$, but not always. So the statement $1$ is True. For statement-$2L_1$ is a diameter of $C$ then $L_2$ is a chord of $C$ as the distance between $L_1$ and $L_2=\frac{6}{\sqrt{13}}<2$ the radius of the circle $C$. So statement-$2$ is false.
Ans: C