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Q.
If a circle cuts the rectangular hyperbola $xy = 1$ in the points $(x_r,y_r)$ where $r - 1, 2, 3, 4$ then
Conic Sections
Solution:
Let the circle be
$x^{2}+y^{2}+2gx+2fy +K = 0 \quad...\left(1\right) $
and hyperbola be $xy = 1\quad...\left(2\right)$
From $\left(2\right), y=\frac{1}{x}$. Putting in $\left(1\right)$, we get
$x^{2}+\frac{1}{x^{2}} +2gx+\frac{2f}{x}+K=0 $
i.e., $x^{4}+2gx^{3}+Kx^{2}+2fx+1 = 0$
Let its root be $x_{1}, x_{2}, x_{3}, x_{4} $
$\therefore x_{1} x_{2}x_{3}x_{4} = 1$