If a circle intersects the hyperbola xy = 1 a | vedclass

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(B) Let the equation of the circle be $x^2 + y^2 + 2gx + 2fy + c = 0$.Since the circle intersects the hyperbola $xy = 1$,we have $y = \frac{1}{x}$.Sub

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$x_1x_2x_3x_4 = 1$

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(B) Let the equation of the circle be $x^2 + y^2 + 2gx + 2fy + c = 0$.Since the circle intersects the hyperbola $xy = 1$,we have $y = \frac{1}{x}$.Substituting this into the circle equation: $x^2 + (\frac{1}{x})^2 + 2gx + 2f(\frac{1}{x}) + c = 0$.Multiplying by $x^2$,we get the quartic equation: $x^4 + 2gx^3 + cx^2 + 2fx + 1 = 0$.Let the roots of this equation be $x_1, x_2, x_3, x_4$.By Vieta's formulas,the product of the roots is given by the constant term divided by the leading coefficient: $x_1x_2x_3x_4 = \frac{1}{1} = 1$.

If a circle intersects the hyperbola $xy = 1$ at four points $(x_r, y_r)$ for $r = 1, 2, 3, 4$,then:

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