If x-acos t-t sin t and y-asin t-t cos t- find d2 y-d x2- Mention the domain in which it is valid-

Given, x = a(cos t+t sin t) and y = a(sin t t cos t) Differtiating w.r.t. t, we get dx/dt=a[d/dtcos t+(td/dt(sin t)+sin t d/dt(t) ) ] ∴ dx/dt=a{ sin t+(t cos ...

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Byju's Answer

Standard XII

Mathematics

Functions

If x=acos t+t...

Question

$I f x = a (c o s t + t s i n t) a n d y = a (s i n t - t c o s t), f i n d \frac{d^{2} y}{d x^{2}} .$ Mention the domain in which it is valid.

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Solution

Given, x = a(cos t+t sin t) and y = a(sin t-t cos t)

Differtiating w.r.t. t, we get

$\frac{d x}{d t} = a [\frac{d}{d t} c o s t + (t \frac{d}{d t} (s i n t) + s i n t \frac{d}{d t} (t))] ∴ \frac{d x}{d t} = a {- s i n t + (t c o s t + s i n t .1)} = a t c o s t a n d \frac{d y}{d t} = a [\frac{d}{d t} s i n t + (t \frac{d}{d t} c o s t t + c o s t .1)] \Rightarrow \frac{d y}{d t} = a {c o s t - (t (- s i n t) + c o s t .1)} = a t s i n t \Rightarrow \frac{d y}{d x} = \frac{\frac{d y}{d t}}{\frac{d x}{d t}} = \frac{a t s i n t}{a t c o s t} = t a n t i . e ., \frac{d y}{d x} = t a n t . . (i) D i f f e r e n t i a t i n g a g a i n w . r . t . x, w e g e t \frac{d^{2} y}{d x^{2}} = \frac{d}{d x} (t a n t) = \frac{d}{d t} (t a n t) \frac{d t}{d x} = \frac{s e c^{2} t}{a t c o s t} {∵ \frac{d x}{d t} = a t c o s t} = \frac{1}{a t} s e c^{3} t$

The result is valid for all real t except 0 and odd multiples of $\frac{π}{2}$ .

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If x=a(cos t+t sin t) and y=a(sin t−t cos t),find d2ydx2. Mention the domain in which it is valid.

$I f x = a (c o s t + t s i n t) a n d y = a (s i n t - t c o s t), f i n d \frac{d^{2} y}{d x^{2}} .$ Mention the domain in which it is valid.