If ax2+cy+a′x2+c′=0and x is a rational function of y and ac is negative, then

Given, ax2+cy+a′x2+c′=0or   x2ay+a′+cy+c′=0If x is rational, then the discriminant of the above equation must be a perfect square. Hence,              0−4ay+a′cy+c′must be a perfect square⇒ −acy2−ac′+a′cy−a′c′                     must be a perfect square⇒ ac′+a′c2−4ac⋅a′c′=0                    [∵D=0]⇒ ac′−a′c2=0⇒ ac′=a′c⇒ aa′=cc′