If tangents `O Q` and `O R` are dawn to variable circles having radius `r` and the center lying on the rectangular hyperbola `x y=1` , then the locus of the circumcenter of triangle `O Q R` is `(O` being the origin). (a)`x y=4` (b) `x y=1/4` (c)`x y=1` (d) none of these

Options

A. xy = 4
B. xy = 1/4
C. xy = 1
D. none of these

Answer

B

Explanation

Let S be a point on the rectangular hyperbola [say (t,1/t)].
Now, the circumcircle of `Delta`OQR also passes through S.

Therefore, the circumcenter is the midpoint of OS. Hence,
`x=(t)/(2),y=(1)/(2t)`
So, the locus of the circumcenter is `xy=1//4`.