Consider a hyperbola $\mathrm{xy}=4$. Tangent at any point P of hyperbola intersects the coordinate axes at A and B. O is the centre of hyperbola. The locus of the circumcentre of $∆$OAB is

Given hyperbola $\mathrm{xy}=4$ 

Equation of tangent at P(t) is $\mathrm{x}+{\mathrm{yt}}^2-4\mathrm{t}=0$

$\frac{\mathrm{x}}{4\mathrm{t}}+\frac{\mathrm{y}}{\frac{4}{\mathrm{t}}}=1$

It cuts coordinate axes at $\mathrm{A}(4\mathrm{t},0); \mathrm{B}\left(0,\frac{4}{\mathrm{t}}\right)$

Let $\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)$ be the circum centre of $∆$OAB ,

$\therefore$$\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)$ is mid point of AB

$\Rightarrow$ $\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)=\left(2\mathrm{t},\frac{2}{\mathrm{t}}\right)$ 

$\Rightarrow x_1=2t,y_1=\frac{2}{t}$

By  eliminating t

we get $x_1{y}_1=4$

locus is $\mathrm{xy}=4$  which is rectangular hyperbola,

$$\begin{gathered} copare with xy=c^2 \\ \Rightarrow c^2=4 \end{gathered}$$

Now $\text{ L. L. }\mathrm{R}=2\sqrt{2}\mathrm{c}=2\sqrt{2}\left(2\right)=4\sqrt{2}$