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 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

Answer: (C)

$5$

REF.Image

Given : Q is on y axis

$\Rightarrow (x=0)$

$5\left(0\right)-2\left(y\right)+6=0$

$\Rightarrow y=3$

$\therefore Q=(0,3)$

$(OP{)}^2+(PQ{)}^2=(OQ{)}^2$ _____ (1)

$x^2+y^2+6x+6y-2=0$

$0\to (-3,-3)\to$ center of circle

$OQ=\sqrt{(-3-0{)}^2+(-3-3{)}^2}=\sqrt{9+36}\Rightarrow \sqrt{45}$ ___ (2)

$OP=$ radius of circle = $\sqrt{3^2+3^2-(-2)}\Rightarrow \sqrt{18+2}\Rightarrow \sqrt{20}$ ___ (3)

from, (1), (2) & (3)

$(OP{)}^2+(P{Q}^2)=(OQ{)}^2$

$(\sqrt{20}{)}^2+(PQ{)}^2=(\sqrt{45}{)}^2$

$P{Q}^2=45-20=25$

$\Rightarrow PQ=5$