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If $2 x+y+k=0$ is a normal to the parabola $\mathrm{y}^{2}=-8 x$, then the value of $k$, is
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Verified Answer
The correct answer is:
24
The equation of any normal to the parabola
$$
y^{2}=-8 x \text { is } y=m x+4 m+2 m^{3}...(i)
$$
(using equation of normal of parabola in slope form $y=m x-2 a m-a m^{3}$ and $a=-2$ ) The given normal is
$$
2 x+y+k=0 \Rightarrow y=-2 x-k...(ii)
$$
Comparing eqs. (i) and (ii), we get
$$
\begin{array}{l}
m=-2 \text { and }-4 m-2 m^{3}=k \\
\Rightarrow k=8+16=24
\end{array}
$$
$$
y^{2}=-8 x \text { is } y=m x+4 m+2 m^{3}...(i)
$$
(using equation of normal of parabola in slope form $y=m x-2 a m-a m^{3}$ and $a=-2$ ) The given normal is
$$
2 x+y+k=0 \Rightarrow y=-2 x-k...(ii)
$$
Comparing eqs. (i) and (ii), we get
$$
\begin{array}{l}
m=-2 \text { and }-4 m-2 m^{3}=k \\
\Rightarrow k=8+16=24
\end{array}
$$
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