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(B) Let the equation of the circle be $x^2 + y^2 + 2gx + 2fy + k = 0$.The parametric coordinates of any point on the rectangular hyperbola $xy = c^2$

Vedclass · Accepted Answer

$t_1t_2t_3t_4 = 1$

Vedclass · Answer

(B) Let the equation of the circle be $x^2 + y^2 + 2gx + 2fy + k = 0$.The parametric coordinates of any point on the rectangular hyperbola $xy = c^2$ are $(ct, c/t)$.Substituting these into the circle equation:$(ct)^2 + (c/t)^2 + 2g(ct) + 2f(c/t) + k = 0$Multiplying by $t^2$ to clear the denominator:$c^2t^4 + 2gct^3 + kt^2 + 2fct + c^2 = 0$This is a biquadratic equation in $t$ whose roots are $t_1, t_2, t_3,$ and $t_4$.From the properties of roots of a polynomial equation $a_n x^n + a_{n-1} x^{n-1} + \dots + a_0 = 0$,the product of the roots is given by $(-1)^n \frac{a_0}{a_n}$.Here,the product of the roots is $t_1t_2t_3t_4 = (-1)^4 \frac{c^2}{c^2} = 1$.

If a circle cuts a rectangular hyperbola $xy = c^2$ at four points $A, B, C,$ and $D$,and the parameters of these four points are $t_1, t_2, t_3,$ and $t_4$ respectively,then which of the following is true?

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