Find $\frac{d y}{d x}$ , if x and y are connected parametrically by the equations given in questions without eliminating the parameter.

$x = 2 a t^{2}, y = a t^{4} .$

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Solution

Given, $x = 2 a t^{2}, y = a t^{4}$

Differentiating w.r.t. t, we get

$\frac{d x}{d t} = (2 a) (2 t) a n d \frac{d y}{d t} = a (4 t^{3}) ∴ \frac{d y}{d x} = \frac{\frac{d y}{d t}}{\frac{d x}{d t}} = \frac{d y}{d t} \times \frac{d t}{d x} (∵ \frac{d y}{d x} = \frac{d y / d t}{d x / d t}) = \frac{4 a t^{3}}{4 a t} = \frac{t^{3}}{t} = t^{2}$

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