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If the tangent at the point $({at}^{2}, {at}^{3})$ on the curve ${ay}^{2} = x^{3}$ meets the curve again at

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$(\frac{{at}^{2}}{4}, 8 at)$

$({at}^{2}, 2 at)$

$(\frac{{at}^{2}}{4}, \frac{{at}^{3}}{8})$

$(\frac{{at}^{2}}{2}, 2 {at}^{2})$

answer is A.

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Detailed Solution

$\begin{array}{rcl} a y^{2 = x^{3}} & \Rightarrow & \frac{dy}{dx} = \frac{3 x^{2}}{2 ay} = \frac{3 (a^{2} t 4)}{2 a^{2} t^{3}} \\ = & \frac{3 t}{2} \end{array}$

eq of Tangent

$3 t x - 2 y = {at}^{3}$

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