If the tangent drawn at the point $P$ on the circle $x^2+y^2$ $+6 x+6 y=2$ meets the straight line $5 x-2 y+6=0$ at a point $\mathrm{Q}$ on the Y-axis, then the length of $\mathrm{PQ}$ is

• Correct answer
  A

  $5$
• B

  $4$
• C

  $2$
• D

  $1$

Solution

$\because Q$ lies on $y$-axis $\begin{aligned} & \therefore \quad x=0 \\ & \Rightarrow-2 y+6=0 \Rightarrow y=3 \end{aligned}$
$\begin{aligned} \therefore \quad & Q(0,3) \\ & \text { also } P Q=\text { Length of tangent } \\ & O Q=\sqrt{(-3-0)^2+(-3-3)^2}=\sqrt{45} \\ & O P=\text { radius }=\sqrt{3^2+3^2+2}=\sqrt{20} \\ \therefore & O Q^2=O P^2+P Q^2 \\ \Rightarrow & 45=20+P Q^2 \Rightarrow P Q^2=25 \Rightarrow P Q=5 .\end{aligned}$

Hence, the correct answer is option: 'A'.