Q. A circle cuts rectangular hyperbola $xy = 1$ in the points $\left( x_{r},\ y_{r} \right),\ r = 1,2,\ 3,\ 4,$ then
A) $y_{1}y_{2}y_{3}y_{4} = 1$
B) $x_{1}x_{2}x_{3}x_{4} = 1$
C) $x_{1}x_{2}x_{3}x_{4} = y_{1}y_{2}y_{3}y_{4} = - 1$
D) $y_{1}y_{2}y_{3}y_{4} = 0$
Explanation
(a)
Let the equation of circle is
$x^{2} + y^{2} + 2gx + 2\ fy + c = 0$ …(i)
Given that, $xy = 1$ …(ii)
From Eq.(i) and (ii), we get
$x^{4} + 2gx^{3} + cx^{2} + 2fx + 1 = 0$
$\therefore$ Product of roots $x_{1}x_{2}x_{3}x_{4} = 1$ and similarly
$y_{1}y_{2}y_{3}y_{4} = 1$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.