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Equation of Normal at Given Point

Let x = my + ...

Question

Let $x = m y + c$ is normal to $x^{2} = 4 y$ , if $k^{2} + m k + m = 0$ has only one real value of $k$ ,then value(s) of $c$ is/are

0

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72

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- 72

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64

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Solution

The correct options are
A $0$
C $- 72$
Using the slope form of the normal for the given parabola, we get
$x = m y - 2 m - m^{3}$
$∴ c = - 2 m - m^{3}$
Now $k^{2} + m k + m = 0$ will have equal roots if
$m^{2} - 4 m = 0 \Rightarrow m = 0, 4$
Hence values of $c = 0, - 72$

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