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From a point ...

Question

From a point $(C, 0)$ three normals are drawn to the parabola $y^{2} = x$ . Then

C < \frac{1}{2}

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C = \frac{1}{2}

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C > \frac{1}{2}

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\frac{1}{2} > C > \frac{1}{4}

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Solution

The correct option is C $C > \frac{1}{2}$
equation of normal to the parabola at point $(a m^{2}, - 2 a m)$ is $y = m x - 2 a m - a m^{3}$

satisfying the point $(C, 0)$ in the equation

$0 = C m - 2 a m - a m^{3}$

$m (a m^{2} - (C - 2 a)) = 0$
and here $a = \frac{1}{4}$

So, $C - 2 a > 0$

$C$ must be $C > \frac{1}{2}$

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