Question

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. $t_{3} = t_{4}$

Answer 1

Answer: (B)

$t_{1}t_{2}t_{3}t_{4} = 1$

Answer 2

Let the equation of circle be $x^{2} + y^{2} = a^{2}$ ......(i)

Parametric equation of rectangular hyperbola is $x = ct,y = \frac{c}{t}$

Put the values of x and y in equation (i) we get $c^{2}t^{2} + \frac{c^{2}}{t^{2}} = a^{2}$⇒$c^{2}t^{4} - a^{2}t^{2} + c^{2} = 0$

Hence product of roots$t_{1}t_{2}t_{3}t_{4} = \frac{c^{2}}{c^{2}} = 1$