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Q.
If the normal to the rectangular hyperbola xy = c$^2$ at the point $\bigg(ct, \frac{c}{t}\bigg)$ meets the curve again at $\bigg(ct', \frac{c}{t'}\bigg),$then
The equation of tangent at $\bigg(ct, \frac{c}{t}\bigg)$ is
ty= $t^3 x-ct^4+c$
if it passes through $\bigg(ct' , \frac{c}{t'}\bigg)$then
$\Rightarrow \, \, \, \frac{tc}{t'}=t^3 ct' -ct^4 + c$
$\Rightarrow \, \, t=t^3 t^{'2}-t^4 t' +t'$
$\Rightarrow \, \, \, t . t' \, = \, t^3t'(t' .t) \Rightarrow \, \, t^3t'= -1$
Note : If we take the co-ordinate axes along the asymptotes of a rectangular hyperbola, then the general equation $x^2 - y^2 =a^2$ becomes xy = $c^2, $ where c is a constant.