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  • What is the product of parameters of four concyclic points on the hyperbola .

What is the product of parameters of four concyclic points on the hyperbola \mathrm{x y=c^2}.
 

Option: 1

\mathrm{ c^4}


Option: 2

1


Option: 3

4


Option: 4

None of these


Answers (1)

best_answer

Given equation of hyperbola is \mathrm{x y=c^2}

Let four concyclic points on the hyperbola be \mathrm{\left(x_n, y_i\right); i=1,2,3,4.}

Let the equation of the circle through points A, B, C and D be

\mathrm{x^2+y^2+2 g x+2 f y+d=0 }

Solving circle and hyperbola, we get

\begin{aligned} &\mathrm{ x^2+\frac{c^4}{x^2}+2 g x+2 f \cdot \frac{c^2}{x}+d=0 }\\ \Rightarrow &\mathrm{ x^4+2 g x^3+d x^2+2 f c^2 x+c^4=0 }\\ \therefore \quad &\mathrm{ \text { Product of roots, } x_1 x_i x_3 x_4=c^4 }\\ &\mathrm{ \text { Now, }\left(x_i, y_i\right)=\left(c t_i, \frac{c}{t_i}\right) }\\ &\mathrm{ \therefore \quad\left(c t_1\right)\left(c t_2\right)\left(c t_3\right)\left(c t_4\right)=c^4 \Rightarrow t_1 t_2 t_3 t_4=1}\\ \end{aligned}

Posted by

Pankaj Sanodiya

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