For different real non-zero numbers x_1, x_2, | vedclass

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(A) Let the equation of the circle be $x^2 + y^2 + 2gx + 2fy + c = 0$. Since the points $(x_i, \frac{1}{x_i})$ lie on the circle for $i = 1, 2, 3, 4$,

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(A) Let the equation of the circle be $x^2 + y^2 + 2gx + 2fy + c = 0$. Since the points $(x_i, \frac{1}{x_i})$ lie on the circle for $i = 1, 2, 3, 4$,we have: $x_i^2 + (\frac{1}{x_i})^2 + 2gx_i + 2f(\frac{1}{x_i}) + c = 0$. Multiplying throughout by $x_i^2$,we get the quartic equation: $x_i^4 + 2gx_i^3 + cx_i^2 + 2fx_i + 1 = 0$. This equation has roots $x_1, x_2, x_3, x_4$. By Vieta's formulas,the product of the roots of a quartic equation $ax^4 + bx^3 + cx^2 + dx + e = 0$ is given by $\frac{e}{a}$. Here,$a = 1$ and $e = 1$. Therefore,$x_1 x_2 x_3 x_4 = \frac{1}{1} = 1$.

For different real non-zero numbers $x_1, x_2, x_3$ and $x_4$,suppose the points $(x_1, \frac{1}{x_1}), (x_2, \frac{1}{x_2}), (x_3, \frac{1}{x_3})$ and $(x_4, \frac{1}{x_4})$ lie on the boundary of a circle of radius $4$. Then,the value of $x_1 x_2 x_3 x_4$ is

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