Question

Let the equation of a curve be x = a (q + sin q), y = a

(1 – cos q). If q changes at a constant rate k then the rate of change of the slope of the tangent to the curve at q = p/3 is

• A. 2k/Ö3
• B. k/Ö3
• C. k
• D. None of these

Answer 1

Answer: (D)

None of these

Answer 2

$\frac{dy}{dx}$ = $\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}$ = $\frac{a\sin\theta}{a(1 + \cos\theta)}$ = tan $\frac{\theta}{2}$,

\ the rate of change of the slope, i.e.,

$\frac{dy}{dt}$ = $\frac{d\left( \frac{dy}{dx} \right)}{dt}$ = $\frac{1}{2}$ sec² $\frac{\theta}{2}$ . $\frac{d\theta}{dt}$ = $\frac{k}{2}$ sec² $\frac{\theta}{2}$

\ the required rate = $\frac{k}{2}$. sec² $\frac{\pi}{6}$ = $\frac{k}{2}$. $\left( \frac{2}{\sqrt{3}} \right)^{2}$.