Q. An object is moving in the clockwise direction around the unit circle $x^{2} + y^{2} = 1.$ As it passes through the point $\left( \frac{1}{2},\frac{\sqrt{3}}{2} \right),$its $y -$coordinate is decreasing at the rate of 3unit per second.The rate at which the$x -$coordinate changes at this point is (in unit per second)
A) 2
B) $3\sqrt{3}$
C) $\sqrt{3}$
D) $2\sqrt{3}$
Explanation
(b)
The equation of given circle is
$x^{2} + y^{2} = 1$
On differentaing w.r.t. $t$, we get
$2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 0$
But we have, $x = \frac{1}{2},y = \frac{\sqrt{3}}{2}$ and $\frac{dy}{dt} = - 3$, then
$\frac{1}{2}\frac{dx}{dt} + \frac{\sqrt{3}}{2}( - 3) = 0\ \ \ \Rightarrow \ \ \ \frac{dx}{dt} = 3\sqrt{3}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.