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  • If a circle cuts a rectangular hyperbola <img alt="\mathrm{x y=c^2 \text { in } A, B, C, D}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Bx%20y%3Dc%5E2%20%5Ctext%20%7B%20in

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

Option: 1

\mathrm{t_1 t_2=t_3 t_4}


Option: 2

\mathrm{t_1 t_2 t_3 t_4=1}


Option: 3

\mathrm{t_1=t_2}


Option: 4

\mathrm{t_3=t_4}


Answers (1)

best_answer

Let the equation of circle be \mathrm{x^2+y^2=a^2\ \ \ .............(i)}
Parametric equation of rectangular hyperbola is \mathrm{x=c t, y=\frac{c}{t}}
Put the values of x and y in equation \mathrm{(i)} we get \mathrm{c^2 t^2+\frac{c^2}{t^2}=a^2 \Rightarrow c^2 t^4-a^2 t^2+c^2=0}
Hence product of roots \mathrm{t_1 t_2 t_3 t_4=\frac{c^2}{c^2}=1}

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Gaurav

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