If a circle and the rectangular hyperbola $x y = c^{2}$ meet in the four points $t_{1} . t_{2} . t_{3} & t_{4}$ then $t_{1} . t_{2} . t_{3} t_{4}$ is equal to______

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Solution

Let the equation of the circle be $a x^{2} + b y^{2} + 2 g x + 2 f y + c = 0$ - - - - - - - (1)

Equation of the rectangular hyperbola $x y = c^{2}$

solving equation (1) & (2)

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

$x^{2} + \frac{c^{4}}{x^{2}} + 2 g x + 2 f \frac{c^{2}}{x} + c = 0$

$x^{4} + 2 g x^{3} + c x^{2} + 2 f c^{2} x + c^{4} = 0$

then product of the roots of the above equation.

$x_{1} . x_{2} . x_{3} . x_{4} = \frac{c^{4}}{1}$

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

$c^{4} . t_{1} . t_{2} . t_{3} t_{4} = c 4$

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

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