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Q.

At any point for the curve $3 y^{2} = {(x + 5)}^{3}$ , If ST represents the length of the subtangent and SN represents the length of the subnormal then $9 {(ST)}^{2} =$

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a

$27 (SN)$

b

$8 (SN)$

c

$\frac{8}{3} (S N)$

d

$8 {(S N)}^{2}$

answer is D.

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Detailed Solution

$G i v e n c u r v e 3 y^{2} = {(x + 5)}^{3}; p (x, y) \begin{matrix} ∴ \frac{d y}{d x} = \frac{{(x + 5)}^{2}}{2 y} = m \\ ∴ S N = |y . m| = \frac{1}{2} {(x + 5)}^{2} \\ 9 {(S . T .)}^{2} = 9 {(\frac{y}{m})}^{2} \\ = 9 \frac{y^{2}}{{(\frac{{(x + 5)}^{2}}{2 y})}^{2}} = \frac{36 y^{4}}{{(x + 5)}^{4}} = \frac{4 {(x + 5)}^{6}}{{(x + 5)}^{4}} = 4 {(x + 5)}^{2} = 8 (S N) \end{matrix}$

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At any point for the curve 3y2=(x+5)3, If ST represents the length of the subtangent and SN represents the length of the subnormal then 9(ST)2=