Question

If $2x+y+k=0$ is a normal to the parabola $y^2=-8x$, then the value of $k$ is

Answer 1

The equation of any normal to $y^2=-8x$ is
$y=mx+4m+2m^3\cdots \left(i\right)$
[Using $y=mx-2am-a{m}^3$]

The given line is $2x+y+k=0$
$\Rightarrow y=-2x-k\cdots \left(ii\right)$
On comparing equations $\left(i\right)$ and $\left(ii\right)$, we get
$m=-2\text{and}k=-4m-2m^3\therefore k=8+16=24$