If tangent and normal to a rectangular hyperbola xy = c^2 at a variable point cut off intercept

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10-2. Parabola, Ellipse, Hyperbola

normal

If the tangent and normal to a rectangular hyperbola $xy = c^2$ at a variable point cut off intercept $a_1, a_2$ on $x-$ axis and $b_1, b_2$ on $y-$ axis, then $(a_1a_2 + b_1b_2)$ is

A

$2$

B

$\frac {1}{2}$

C

$0$

D

$-1$

Solution

$x y=c^{2}$

$2 x y=2 c^{2}$

$\frac{\mathrm{xc}}{\mathrm{t}}+(\mathrm{ct}) \mathrm{y}=2 \mathrm{c}^{2}(\text { tangent })$

$\frac{\mathrm{x}}{\mathrm{t}}+\mathrm{ty}=2 \mathrm{c}$

$a_{1}=2 c t, \quad b_{1}=\frac{2 c}{t}$

for normal

$ y-\frac{c}{t} =t^{2}(x-c t) $

${a_2} = \left( {ct – \frac{c}{{{t^3}}}} \right)$ and ${b_2} = \left( {\frac{c}{t} – c{t^3}} \right)$

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

Standard 11

Mathematics

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