Question

If x = a(cos t + t sin t) and y = a(sin t – t cos t), then `(d^2y)/(dx^2)` is ______.

विकल्प

• sec³ t

• at sec³ t

• `(sec^3t)/(at)`

• sec² t

Answer 1

If x = a(cos t + t sin t) and y = a(sin t – t cos t), then `(d^2y)/(dx^2)` is `underlinebb((sec^3t)/(at))`.

Explanation:

It is given that x = a(cos t + t sin t) and y = (sin t – t cost).

Therefore, `dx/dt` = a[–sin t + sin t + t cos t] = at cos t

`dy/dt` = a[cos t – {cos t – t sin t}] = at sin t

∴ `dy/dx = ((dy/dt))/((dx/dt)) = (at sin t)/(at cos t)` = tan t

Then, `(d^2y)/(dx^2) = d/dx(dy/dx)`

= `d/dx(tan t)`

= `d/dt(tan t)dt/dx`

= `sec^2t. dt/dx`

= `sec^2t. 1/(at cos t)`

= `(sec^3t)/(at)`