Question

lf 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. $4$
• B. $2\sqrt{5}$
• C. $5$
• D. $3\sqrt{5}$

Answer 1

The correct option is C $5$

$x^2+y^2+6x+6y-2=0$
Centre $(-3,-3)$, radius $=\sqrt{20}$,$Q=(0,3)$
Tangent PQ is drawn to circle $x^2+y^2+6x+6y-2=0$
$\therefore$ In CPQ,
$C{P}^2+P{Q}^2=C{Q}^2$
$r^2+P{Q}^2=(\sqrt{3^2+6^2}{)}^2$
$\Rightarrow 20+P{Q}^2=(\sqrt{45}{)}^2$
$\Rightarrow P{Q}^2=45-20$
$\Rightarrow PQ=\sqrt{25}$
$\Rightarrow PQ=5$