The condition that the two tangents to the parabola $x^{2} = 4 a y$ become normals to the circle $x^{2} + y^{2} - 2 a x - 2 b y - c = 0$ is given by

a^{2} = 4 b

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a^{2} > 4 a b

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a^{2} > 4 b

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a > 4 b

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Solution

The correct option is B $a^{2} > 4 a b$
Tangents to parabola are $x = m y + \frac{a}{m}$
they have to pass through $(a, b)$ for it to be a normal
$\Rightarrow a = m b + \frac{a}{m}$
$\Rightarrow m^{2} b - a m + a = 0$
$Δ > 0$ for two values of m
$\Rightarrow a^{2} - 4 a b > 0$
$a^{2} > 4 a b$

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