If x = a cos θ,y=b sin θ,then d3ydx3 is equal to

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Question

If x = a

c o s θ, y = b s i n θ, t h e n \frac{d^{3} y}{d x^{3}}

is equal to

Options:

A . (−3ba3)cosec4θ cot4θ

B . (3ba3)cosec4θ cotθ

C . (−3ba3)cosec4θ cotθ

D . None of the above

Answer: Option C
:
C

∵ x = a c o s θ \Rightarrow \frac{d x}{d θ} = - a s i n θ a n d y = b s i n θ \Rightarrow \frac{d y}{d θ} = b c o s θ ∴ \frac{d y}{d x} = - \frac{b}{a} c o t θ \Rightarrow \frac{d^{2} y}{d x^{2}} = \frac{b}{a} c o s e c^{2} θ \frac{d θ}{d x} = - \frac{b}{a^{2}} c o s e c^{3} θ ∴ \frac{d^{3} y}{d x^{3}} = \frac{3 b}{a^{2}} c o s e c^{2} θ (- c o s e c θ c o t θ) \frac{d θ}{d x} = \frac{3 b}{a^{2}} c o s e c^{3} θ c o t θ (- \frac{1}{a s i n θ}) = - \frac{3 b}{a^{3}} c o s e c^{4} θ c o t θ

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