Question

A stick of length $r$ units slides with its ends on coordinate axes. Then the locus of the midpoint of the stick is a curve whose length is

• A. $2\pi$
• B. $\pi^2$
• C. $\dfrac{1}{2}\pi$
• D. $\pi$

Hint:

When endpoints trace coordinate axes with fixed distance, the midpoint traces a quarter circle.

Answer 1

The Correct Option is D

Let the endpoints of the stick be $(x,0)$ and $(0,y)$ such that $x^2 + y^2 = r^2$.
The midpoint is $\left(\dfrac{x}{2}, \dfrac{y}{2}\right)$ and it satisfies $\left(\dfrac{x}{2}\right)^2 + \left(\dfrac{y}{2}\right)^2 = \dfrac{r^2}{4}$
So the midpoint moves along a quadrant of a circle of radius $\dfrac{r}{2}$.
Length of a quadrant = $\dfrac{1}{4} \cdot 2\pi \cdot \dfrac{r}{2} = \dfrac{\pi r}{4}$
For $r = 2$, the required length = $\pi$