Question

A line segment of 8 units in length moves so that its end points are always on the coordinate axes. Then, the equation of locus of its mid-point is

• A. $x^2+y^2=4$
• B. $x^2+y^2=16$
• C. $x^2+y^2=8$
• D. $|x|+|y|=8$

Answer 1

Answer: (B)

Let the line segment whose length is 8 unit intercept the coordinate axes at
$A(a,0)$ and $B(0,b)$
x-axes and y-axes respectively.
Let $P(h,k)$ be the mid-point of line segment AB.
$\Rightarrow$ $h=a/2$ and $k=b/2$
$\Rightarrow$ $a=2h$ and $b=2k$ ..(i)
Now, $AB=\sqrt{{(a-0)}^2+{(b-0)}^2}$
$A{B}^2=a^2+b^2$
$\Rightarrow$ ${(8)}^2={(2h)}^2+{(2k)}^2$
(from Eq. (i)] $\Rightarrow$ $h^2+k^2=16$
$\therefore$ The leocus of mid-point P is
$x^2+y^2=16$