The equation of the tangent at the point $P (t),$ where $t$ is any parameter, to the parabola $y^{2} = 4 a x$ is

y t = x + a t^{2}

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y = x t + a t^{2}

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y = t x

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

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Solution

The correct option is A $y t = x + a t^{2}$
$P (t) = (a t^{2}, 2 a t)$
Slope of tangent at P to parabola $y^{2} = 4 a x$ is $m = \frac{2 a}{y} = \frac{1}{t}$
Therefore required equation is
$(y - 2 a t) = \frac{1}{t} (x - a t^{2})$
$\Rightarrow t y = x + a t^{2}$

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y + 2 a t = t (x - a t^{2})

t

x = a t^{2}, y = 2

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