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The points $A$ and $B,$ whose $x$ coordinates are $a$ and $b$ respectively, lie on the curve

y = \frac{1}{x}

The point $C$ is the point of intersection of the tangents to the curve at $A$ and $B$ .

Given that $a b = - 1,$ prove that $C$ lies on the line $y = - x$

First of all, because $a b = - 1$ we have $b = - \frac{1}{a}$

Because $y = \frac{1}{x},$ you can now find the coordinates of both $A$ and $B$ in terms of $a$ only.

You should get

\begin{aligned} A & = (a, \frac{1}{a}) \\ B & = (- \frac{1}{a}, - a) \end{aligned}