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Parametric Equation of Tangent

Prove that th...

Question

Prove that the tangent at any point to the rectangular hyperbola forms with the asymptotes, a triangle of constant area.

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Solution

The equation of tangent to $x y = c^{2}$ at ' $t$ ' is
$x + y t^{2} = 2 c t$
When $y = 0$ ; $x = 2 c t$
$∴ Q = (2 c t, 0)$
When $x = 0$ ; $y = \frac{2 c t}{t^{2}} = \frac{2 c}{t}$
$∴ R = (0, \frac{2 c}{t})$
Now area of $Δ O Q R$ $= \frac{1}{2} \times 2 c t \times \frac{2 c}{t}$
$= 2 c^{2}$ , which is a constant.

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