Equation of tangent for the parabola $x^{2} = 4 a y$ is

y = m x - a m^{2}

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y = m x + a m^{2}

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y = m x - \frac{a}{m}

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y = m x + \frac{a}{m}

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Solution

The correct option is A $y = m x - a m^{2}$
let tangent is drawn at $(h, k)$ and it has a slope $m$

$x^{2} = 4 a y 2 x = 4 a \frac{d y}{d x} \frac{d y}{d x} = \frac{2 x}{4 a} = \frac{x}{2 a} {(\frac{d y}{d x})}_{(h, k)} = \frac{h}{2 a} = m \Rightarrow h = 2 a m h^{2} = 4 a k \Rightarrow k = \frac{4 a^{2} m^{2}}{4 a} = a m^{2}$

Equation of tangent is

$y - k = m (x - h) y - a m^{2} = m (x - 2 a m) y - a m^{2} = m x - 2 a m^{2} y = m x - a m^{2}$

So option $A$ is correct.

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