Q. If $\left( ax^{2} + c \right)y + \left( d\ x^{2} + c^{'} \right) = 0\ and\ x$ is a rational function of $y\ and\ ac$ is negative, then
A) $ac^{'} + a^{'}c = 0$
B) $\frac{a}{a^{'}} = \frac{c}{c^{'}}$
C) $a^{2} + c^{2} = {a'}^{2} + c^{'2}$
D) $aa' + cc^{'} = 1$
Explanation
(b)
Given, $\left( ax^{2} + c \right)y + \left( a'\ x^{2} + c^{'} \right) = 0$
or $x^{2}\left( ay + a^{'} \right) + \left( cy + c^{'} \right) = 0$
Since, $x$ is rational, then the discriminant of the above equation must be a perfect square.
$\therefore 0 - 4$ $\left( ay + a^{'} \right) + \left( cy + c^{'} \right) = 0$
$\Rightarrow \ - acy^{2} - \left( ac^{'} + a^{'}c \right)y - a^{'}c'$
Must be a perfect square
$\Rightarrow \ \left( ac^{'} - a^{'}c \right)^{2} - 4aca^{'}\ c^{'} = 0$
$\Rightarrow \ \left( ac^{'} - a^{'}c \right)^{2} = 0$
$\Rightarrow ac^{'} = a^{'}c$
$\Rightarrow \frac{a}{a^{'}} = \frac{c}{c^{'}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.