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  4. Normal drawn to the hyperbola x y=1 at point A(t1), where it meets the hyperbola again at B(t2), then

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Q. Normal drawn to the hyperbola $x y=1$ at point $A\left(t_1\right)$, where it meets the hyperbola again at $B\left(t_2\right)$, then

Conic Sections

Solution:

$y =\frac{1}{ x }$
$\frac{ dy }{ dx }=\frac{-1}{ x ^2}$
Equation of normal $A \left( t _1\right)$ and $A \left( t _2\right)$
$\frac{\frac{1}{t_2}-\frac{1}{t_1}}{t_2-t_1}=t_1^2 $
$\Rightarrow t_1^3 t_2=-1$