Find the length of subtangent to $y^{2} = 4 a x$ at $(a t^{2}, 2 a t)$

a t

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2 a t

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2 a t^{2}

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a t^{2}

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Solution

The correct option is D $2 a t^{2}$
The length of sub-tangent to parabola is $y \frac{d x}{d y}$
Given equation of parabola is $y^{2} = 4 a x$
Now differentiate it w.r.t $y$ , we get $\frac{d x}{d y} = \frac{y}{2 a}$
Length of sub-tangent is $\frac{y^{2}}{2 a} = \frac{4 a x}{2 a} = 2 x = 2 a t^{2}$
Therefore option $C$ is correct

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