Question

The mirror image of the directrix of the parabola
$y=x^2+5x+7$ in the line mirror $x+y=1$ is

• A. $2y=1$
• B. $2x-y=1$
• C. $4x=y$
• D. $x=\frac{1}{2}$

Answer 1

The correct option is D $x=\frac{1}{2}$

Equation of directrix of $y=x^2+5x+7$ will be $y=\frac{1}{2}$
Any point on the line $y=\frac{1}{2}$ will be of the form $(k,\frac{1}{2})$ where $k\in R$
Let $(x,y)$ be the locus of all the points which are the mirror image of the points $(k,\frac{1}{2})$ in the line $x+y=1$
$\Rightarrow \frac{x-k}{1}=\frac{y-\frac{1}{2}}{1}=\frac{-2(k+\frac{1}{2}-1)}{2}\Rightarrow x=\frac{1}{2},y=1-k$
$\therefore$ Mirror image of the directrix is the line $x=\frac{1}{2}$