If d2xdy2(dydx)3+d2ydx2=k, then k is equal to

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If $\frac{d^{2} x}{d y^{2}} {(\frac{d y}{d x})}^{3} + \frac{d^{2} y}{d x^{2}} = k$ , then k is equal to

Options:

A . 0

B . 1

C . 2

D . None of these

Answer: Option A
:
A

\frac{d y}{d x} = {(\frac{d x}{d y})}^{- 1} \Rightarrow \frac{d^{2} y}{d x^{2}} = - {(\frac{d x}{d y})}^{- 2} {\frac{d}{d x} (\frac{d x}{d y})} \Rightarrow \frac{d^{2} y}{d x^{2}} = - {(\frac{d x}{d y})}^{- 2} {\frac{d}{d y} (\frac{d x}{d y}) \frac{d y}{d x}} \Rightarrow \frac{d^{2} y}{d x^{2}} = - {(\frac{d y}{d x})}^{2} {\frac{d^{2} x}{d y^{2}} . \frac{d y}{d x}} \Rightarrow \frac{d^{2} y}{d x^{2}} = - {(\frac{d y}{d x})}^{3} \frac{d^{2} x}{d y^{2}} \Rightarrow \frac{d^{2} y}{d x^{2}} + {(\frac{d y}{d x})}^{3} \frac{d^{2} x}{d y^{2}} = 0

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