If ax2+cy+a′x2+c′=0 and 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 equationmust be a perfect square. Hence,0−4ay+a′cy+c′ must be a perfect square⇒ −acy2−ac′+a′cy−a′c′⇒ ac′+a′c2−4aca′c′=0            [∵D=0]⇒ ac′−a′c2=0⇒ ac′=a′c⇒ aa′=cc′