Let a circle C of radius 5 lie below x -axis. line L_{1}=4 x+3 y-2 passes through centre P of circle

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10-1.Circle and System of Circles

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Let a circle $C$ of radius $5$ lie below the $x$-axis. The line $L_{1}=4 x+3 y-2$ passes through the centre $P$ of the circle $C$ and intersects the line $L _{2}: 3 x -4 y -11=0$ at $Q$. The line $L _{2}$ touches $C$ at the point $Q$. Then the distance of $P$ from the line $5 x-12 y+51=0$ is

A

$9$

B

$10$

C

$11$

D

$12$

(JEE MAIN-2022)

Solution

$4 x+3 y+2=0$

$3 x-4 y-11=0$

$\frac{x}{-25}=\frac{y}{50}=\frac{1}{-25}$

$\frac{x-1}{\cos \theta}=\frac{y+2}{\sin \theta}=\pm 5$

$y=-2+5\left(-\frac{4}{5}\right)=-6$

$x=1+5\left(\frac{3}{5}\right)=4$

Req. distance

$\left|\frac{5(4)-12(-6)+51}{13}\right|$

$=\left|\frac{20+72+51}{13}\right|$

$=\frac{143}{13}=11$

Standard 11

Mathematics

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