Question

A line segment of length $l$ slides with its endpoints always on X-axis and Y-axis respectively. The locus of its midpoint is:

• A. $x^2+y^2=l^2$
• B. $x^2+y^2=4l^2$
• C. $x^2+y^2=\frac{l^2}{4}$
• D. $x^2-y^2=l^2$

Answer 1

The correct option is C $x^2+y^2=\frac{l^2}{4}$

Let $(h,k)$ be the mid point of the line. Also let $(a,0)$ and (0,b)$ be the end points of the line.

Then by the problem,

$a^2+b^2=l^2$......(1).

Now the mid point of the line be $(\frac{a}{2},\frac{b}{2})$.

Then $\frac{a}{2}=h$ and $\frac{b}{2}=k$

or, $a=2h$ and $b=2k$.

Putting this values of $a,b$ in (1) we get,

$4(h^2+k^2)=l^2$

or, $h^2+k^2=\frac{l^2}{4}$.

So locus of $(h,k)$ is $x^2+y^2=\frac{l^2}{4}$.