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If the tangent to the curve $2 y^{3} = a x^{2} + x^{3}$ at the point (a, a) cuts off intercepts $α, β$ on co-ordinate axes, where $α^{2} + β^{2} = 61,$ then the value of ‘ $a$ ’ is equal to

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answer is C.

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Detailed Solution

$2 y^{3} = a x^{2} + x^{3}$

$6 y^{2} \frac{d y}{d x} = 2 a x + 3 x^{2}$

$\frac{d y}{d x} = \frac{5}{6}$

$5 x - 6 y + a = 0$

$∴ \frac{a^{2}}{25} + \frac{a^{2}}{36} = 61$

$a = 30$

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