Question

If the angle made by the tangent at the point $\left(x_{0}, y_{0}\right)$ on the curve $x=12(t+\sin t \cos t)$, $y =12(1+\sin t )^{2}, 0< t <\frac{\pi}{2}$, with the positive $x$-axis is $\frac{\pi}{3}$, then $y _{0}$ is equal to

• A. $6(3+2 \sqrt{2})$
• B. $3(7+4 \sqrt{3})$
• C. $27$
• D. $48$

Answer 1

Answer: (C)

$\frac{d y}{d x}=\frac{2(1+\sin t) \times \cos t}{1+\cos 2 t}$
$\Rightarrow \frac{2(1+\sin t) \cos t}{2 \cos ^{2} t}=\sqrt{3}$
$\Rightarrow t=\frac{\pi}{6}, y_{0}=27$