Question

lf $x=2sint-\sin 2t,y=2cost-\cos 2t$, then the value of $\frac{d^{2}y}{d{x}^2}$ at $t=\frac{\pi }{2}$ is

• A. $2$
• B. $-\frac{1}{2}$
• C. $-\frac{3}{4}$
• D. $-\frac{3}{2}$

Answer 1

The correct option is D $-\frac{3}{2}$

$\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{-2sint+2sin2t}{2cost-2cos2t}$
$\frac{d}{dt}\left(\frac{dy}{dx}\right)\frac{dt}{dx}=\frac{d^{2}y}{d{x}^2}=\frac{d}{dt}\left(\frac{sin2t-sint}{cost-cos2t}\right)\frac{1}{(2cost-2cost)}$
${\left(\frac{d^{2}y}{d{x}^2}\right)}_{t=\frac{\pi }{2}}=\left(\frac{(2cos2t-cost)(cost-cos2t)+(sin2t-sint)(sint-2sin2t)}{(cost-cos2t{)}^2(2cost-2cos2t)}\right)$
$=\frac{(-2)\left(1\right)+(-1)\left(1\right)}{2}$
$\frac{-3}{2}$