Question

Assertion :$C:x^2+y^2+6x+6y=2$
$L:5x-2y+6=0$

If the tangent at the point $P$ on the circle $C$ meets the straight line $L$ at a point $Q$ on the $y$-axis then length of $PQ=5.$

Reason: Equation of the chord of contact of $Q$ with respect to the circle $C$ is $3x+6y+7=0$, where $Q$ is the point where the line $L$ meets the axis of $y$.

• 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 B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

$c:x^2+y^2+6x+6y=2$

$L:5x-2y+6=0$

Slope tangent of $c\Rightarrow 32x+2y\frac{dy}{dx}+6+\frac{6dy}{dx}=0$

$\frac{dy}{dx}=\frac{-(2x+6)}{(2y+6)}$ ...(1)

This tangent meets line $5x-2y+6=0$ at $(0,k)$

$-2y+6=0$

$-2k+6=0$

$-2k=-6$

$-k=3$

From eq (1), $\frac{y-3}{x}=\frac{-(2x+6)}{(2y+6)}$

This will give, $PQ=5$

And Reason is also true

Both Assertion and Reason are correct but Reason is not the correct explanation for assertion.