If a circle cuts a rectangular hyperbola xy = {c^2} in A, B, C, D and parameters of these four point

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10-2. Parabola, Ellipse, Hyperbola

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If a circle cuts a rectangular hyperbola $xy = {c^2}$ in $A, B, C, D$ and the parameters of these four points be ${t_1},\;{t_2},\;{t_3}$ and ${t_4}$ respectively. Then

A

${t_1}{t_2} = {t_3}{t_4}$

B

${t_1}{t_2}{t_3}{t_4} = 1$

C

${t_1} = {t_2}$

D

${t_3} = {t_4}$

Solution

(b) Let equation of circle is ${x^2} + {y^2} = {a^2}$

Parametric form of $xy = {c^2}$ are $x = ct,\,\,\,y = \frac{c}{t}$

==> ${c^2}{t^2} + \frac{{{c^2}}}{{{t^2}}} = {a^2}$

==> ${c^2}{t^4} – {a^2}{t^2} + {c^2} = 0$

Product of roots will be, ${t_1}{t_2}{t_3}{t_4} = \frac{{{c^2}}}{{{c^2}}} = 1$.

Standard 11

Mathematics

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