Question

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

• A. 5
• B. 6
• C. 4
• D. 3

Hint:

Use geometry and substitution to find point of intersection and calculate segment length accurately.

Answer 1

The Correct Option is A

Complete square of the circle: $(x+3)^2 + (y+3)^2 = 20$
So, center = $(-3, -3)$ and radius = $\sqrt{20}$
Let point $P$ be any point on the circle, the tangent at $P$ intersects the line $5x - 2y + 6 = 0$ at $Q$ on y-axis
Find intersection point $Q$ by solving tangent and line equations, and compute distance $PQ$
It simplifies to $PQ = 5$ as per calculation