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. \(2\pi r\)
• B. \(\pi r^2\)
• C. \(\dfrac{\pi r}{2}\)
• D. \(\pi r\)

Hint:

For a rod of fixed length sliding between the coordinate axes, the midpoint always traces a circle whose radius is half the length of the rod.

Answer 1

The Correct Option is D

Step 1: Let the coordinates of the endpoints of the stick be determined.
Suppose the stick touches the coordinate axes at the points \[ (a,0) \quad \text{and} \quad (0,b) \] Since the length of the stick is \(r\), using the distance formula we get \[ \sqrt{a^2+b^2}=r \] Squaring both sides, \[ a^2+b^2=r^2 \]

Step 2: Find the midpoint of the stick.
The midpoint of the stick is \[ \left(\frac{a}{2},\frac{b}{2}\right) \] Let the midpoint be \((x,y)\). Then \[ x=\frac{a}{2}, \qquad y=\frac{b}{2} \] Therefore, \[ a=2x, \qquad b=2y \] Substituting these values into \[ a^2+b^2=r^2 \] we get \[ (2x)^2+(2y)^2=r^2 \] \[ 4x^2+4y^2=r^2 \] \[ x^2+y^2=\frac{r^2}{4} \] Thus, the locus of the midpoint is a circle centered at the origin with radius \[ \frac{r}{2} \]

Step 3: Find the length of the locus.
The circumference of a circle of radius \(\dfrac{r}{2}\) is \[ 2\pi \left(\frac{r}{2}\right) \] \[ =\pi r \] Hence, the required length of the curve is \[ \pi r \]

Step 4: Final conclusion.
Therefore, the required answer is \[ \boxed{\pi r} \]