Question

If tangents $OQ$ and $OR$ are drawn to variable circles having radius $r$ and the centre lying on the rectangular hyperbola $xy=1$, then locus of circumcentre of triangle $OQR$ is $(O$ being the origin$)$

• A. $xy=4$
• B. $xy=\frac{1}{4}$
• C. $xy=1$
• D. None of these

Answer 1

The correct option is B $xy=\frac{1}{4}$

Let S $(t,\frac{1}{t})$ be a point on the rectangular hyperbola
Now, circumcircle of $\Delta OQR$ also passes through S.
Therefore, circumcentre is the midpoint of OS. Hence,
$x=\frac{t}{2},y=\frac{1}{2t}$
So, the locus of the circumcentre is $xy=\frac{1}{4}$