Question

A variable triangle $ABC$ is inscribed in a circle of diameter $x$ units. If at a particular instant the rate of change of side, '$a$' is $x/2$ times the rate of change of the opposite angle $A$, then $A$ =

• A. $\pi /6$
• B. $\pi /3$
• C. $\pi /4$
• D. $\pi /2$

Answer 1

Answer: (B)

$\pi /3$

Given $\frac{da}{dt}=\frac{x}{2}\frac{dA}{dt}$

Since, $a=2R\sin A$

$\Rightarrow a=x\sin A$

$\frac{da}{dt}=x\cos A\frac{dA}{dt}$

$\Rightarrow x(1-2\cos A)\frac{dA}{dt}=0$

$\Rightarrow \cos A=\frac{1}{2}$

$\Rightarrow A=\frac{\pi }{3}$