If the line $y=2 x+k$ is normal to the parabola $y^2=4 x$, then $k=$
If the line $y=2 x+k$ is normal to the parabola $y^2=4 x$, then $k=$
-10
10
12
–12
Solution
Since, equation of normal to the parabola $y^2=4 a x$ having slope $m$ is $y=m x-2 a m-a m^3$.
So, equation of normal to the parabola $y^2=4 x$ is
$
y=2 x+k
$
Therefore,
$
\begin{aligned}
k & =-2(1)(2)-(1)(2)^3 \\
& =-4-8=-12
\end{aligned}
$
Hence, option (4) is correct