Question

The mirror image of the directrix of the parabola $y^2=4(x+1)$ in the line mirror $x+2y=3$ is

• A. $4y+3x=16$
• B. $x=-2$
• C. none of these
• D. $3x-4y+16=0$

Answer 1

The correct option is D $3x-4y+16=0$

For the parabola $y^2=4(x+1)$ vertex will be $(-1,0)$ and focus will be $(0,0)$.
Therefore directrix of the parabola will be $x=-2$
Any point on $x=-2\text{is}(-2,k)$
Now, mirror image $(x,y)$ of $(-2,k)$ in the line $x+2y=3$ is given by
$\frac{x+2}{1}=\frac{y-k}{2}=-2\left(\frac{-2+2k-3}{5}\right)\Rightarrow x=\frac{10-4k}{5}-2\Rightarrow x=-\frac{4k}{5}\cdots \left(1\right)\text{Also},y=\frac{20-3k}{5}\cdots \left(2\right)$
From $\left(1\right)$ and $\left(2\right)$ we get,
$y=4+\frac{3}{5}\left(\frac{5x}{4}\right)$
$\therefore 4y=16+3x$ is the equation of the mirror image of the directrix.