Question

A light ray gets reflected from the $x=-2$.If the reflected ray touches the circle $x^2+y^2=4$ and point of incident is $(-2,-4)$,then equation of incident ray is

• A. $3x-4y-10=0$
• B. $3y+4x+20=0$
• C. $4y+2x+20=0$
• D. $y+x+6=0$

Answer 1

The correct option is B $3y+4x+20=0$

Let equation of reflected line with slope $m$ and passes through $(-2,-4)$ is given by,
$(y+4)=m(x+2)\Rightarrow y=mx+2m-4$
Also, reflected ray, touches the circle $x^2+y^2=4$
Thus using condition of tangency, $c^2=a^2(1+m^2)$
$\Rightarrow (2m-4{)}^2=4(1+m^2)\Rightarrow (m-2{)}^2=1+m^2$
$\Rightarrow -4m+3=0\Rightarrow m=\frac{3}{4}$
$m=\frac{-4}{3}$ for incident ray (since they will be perpendicular)

Line passing through $(-2,-4)$ with slope $m=\frac{-4}{3}$ is

$3y+4x+20=0$