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Point Form of Tangent: Hyperbola

If the tangen...

Question

If the tangent and normal to a rectangular hyperbola cut off intercepts $a_{1}$ and $a_{2}$ on one axis and $b_{1}$ and $b_{2}$ on the other, then :

A

a_{1} a_{2} = b_{1} b_{2}

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B

a_{1} a_{2} - b_{1} b_{2} = 1

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C

a_{1} b_{1} + a_{2} b_{2} = 0

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D

a_{1} a_{2} + b_{1} b_{2} = 0

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Solution

The correct option is D $a_{1} a_{2} + b_{1} b_{2} = 0$
Assume rectangular hyperbola is $x y = c^{2}$
Thus equation of tangent and normal at any point 't' are,
$\frac{x}{t} + t y = 2 c$ and $y - \frac{c}{t} = t^{2} (x - c t)$
Now putting $y = 0$ in both the equation we get, $a_{1} = 2 c t, a_{2} = c t - \frac{c}{t^{3}}$
and putting $x = 0$ we get, $b_{1} = \frac{2 c}{t}, b_{2} = \frac{c}{t} - c t^{3}$
$\Rightarrow a_{1} a_{2} + b_{1} b_{2} = 2 c t (c t - \frac{c}{t^{3}}) + \frac{2 c}{t} (\frac{c}{t} - c t^{3}) = 0$
Hence. option 'C' is correct.

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