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. $xy=-4$

Answer 1

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

Let $S$ be any point on the rectangular hyperbola $(t,\frac{1}{t})$
Hence eqaution of the circle with radius $r$ is
$(x-t{)}^2+(y-1/t{)}^2=r^2$
Equation of chord of contact from $(0,0)$ will be
$-xt-\frac{y}{t}+t^2+\frac{1}{t^2}-r^2=0$
Family of circles equation passing through $P,Q$ is
$(x-t{)}^2+(y-1/t{)}^2-r^2+k(-xt-\frac{y}{t}+t^2+\frac{1}{t^2}-r^2)=0$
If it passes through $(0,0)$
$\Rightarrow k=-1$
Hence circumcircle equation for triangle $OQR$ will be
$(x-t{)}^2+(y-1/t{)}^2-r^2+xt+\frac{y}{t}-t^2-\frac{1}{t^2}+r^2=0\Rightarrow x^2+y^2-xt-\frac{y}{t}=0$
whose centre is $(t/2,1/2t)\equiv (h,k)$
$\therefore$ required locus equation will be
$xy=\frac{1}{4}$