Question

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

• A. 8
• B. 16
• C. 24
• D. 32

Answer 1

Answer: (C)

The equation of any normal to the parabola
$y^{2}=-8 x$ is $y=m x+4 m+2 m^{3}$ ....(i)
(using equation of normal of parabola in slope form $y = mx - 2am - am^3$ and $a = - 2)$
But, The given normal is
$2 x+ y +k=0$
$\Rightarrow y=-2 x-k$ ...(ii)
Comparing Eqs. (i) and (ii), we get
$m=-2$
and $-4 m-2 m^{3}=k$
$\Rightarrow k=8+16=24$