Question

Two points $A$ and $B$ move on the $x$-axis and the $y$-axis respectively such that the distance between the two points is always the same. The locus of the middle point of $AB$ is

• A. a straight line
• B. a parabola
• C. a circle
• D. an ellipse

Answer 1

The correct option is C a circle

Let $A=(h,0)$ and $B=(0,k)$
Thus midpoint of $AB$ is $(\frac{h}{2},\frac{k}{2})=P$ (say)
Given length of AB is constant.
$AB=\sqrt{h^2+k^2}=a$ (say), where $a$ is arbitrary constant
Squaring both sides,
$h^2+k^2=a^2\Rightarrow {\left(\frac{h}{2}\right)}^2+{\left(\frac{k}{2}\right)}^2={\left(\frac{a}{2}\right)}^2$
Hence locus of $P(\frac{h}{2},\frac{k}{2})$ is given by, $x^2+y^2={\left(\frac{a}{2}\right)}^2$
Which is clearly a circle.