Question

An object is moving in the clockwise direction around the unit circle $x^2+y^2=1$. As it passes through the point $(\frac{1}{2},\frac{\sqrt{3}}{2})$, its y-coordinate is decreasing at the rate of $3$ unit per second. The rate at which the x-coordinate changes at this point is

• A. $2$
• B. $3\sqrt{3}$
• C. $\sqrt{3}$
• D. $2\sqrt{3}$

Answer 1

The correct option is B $3\sqrt{3}$

The equation of the given circle is $x^2+y^2=1.$

On differentiating w.r.t. $t$, we get

$2x\frac{dx}{dy}+2y\frac{dy}{dx}=0$

But we have $x=\frac{1}{2},y=\frac{\sqrt{3}}{2}$ and $\frac{dy}{dx}=-3$, then

$\frac{1}{2}\frac{dx}{dt}+\frac{\sqrt{3}}{2}(-3)=0\Rightarrow \frac{dx}{dt}=3\sqrt{3}.$