If x = f(t) cos t – f’(t) sin t and y = f(t) sin t + f’(t) cos t, then (dx/dt)^2 + (dy/dt)^2 =

Question

If x = f(t) cos t – f’(t) sin t and y = f(t) sin t + f’(t) cos t, then \((\frac{dx}{dt})^2+(\frac{dy}{dt})^2\)=

A. f(t) – f’’(t) 

B. {f(t) – f’’(t)}² 

C. {f(t) + f’’(t)}² 

D. none of these

Answer 1

Correct Answer is (C) {f(t) + f’’(t)}² 

Given:

x = f(t) cost - f'(t) sint

y = f(t) sint + f'(t) cost

\(\frac{dx}{dt}=f'(t)cost-f(t)sint-f''(t)sint-f'(t)cost\)

= -f(t) sint - f''(t) sint

= -sint[f(t) + f''(t)]

\(\frac{dx}{dt}=f'(t)sint-f(t)cost-f''(t)sint-f'(t)sint\)

= -f(t) cost - f''(t) cost

= cost[f(t) + f''(t)]

= (cos t)² {f(t) + f''(t) ]²

= {f(t) + f’’(t)} 2