Question

Two rods of lengths a and b slides along the x-axis and y-axis respectively in such a manner that their ends are concyclic. The locus of the centre of the circle passing through the end points is

• A.
• B.
• C.
• D.

Answer 1

The correct option is C


Let C(h, k) be the centre of the circle passing through the end points of the rod AB and PQ of lengths a and b respectively. CL and CM be perpendicular from C on AB and PQ respectively.
Then $AL=\left(\frac{1}{2}\right)AB=\frac{a}{2}$
$PM=\left(\frac{1}{2}\right)PQ=\frac{b}{2}$
and CA=CP (radii of the same circle)
$\Rightarrow k^2+\frac{a^2}{4}=h^2+\frac{a^2}{4}\Rightarrow 4(h^2-k^2)=a^2-b^2$
$\therefore$ The locus of (h, k) is $4(x^2-y^2)=a^2-b^2$