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

We have,ax2+cy+a′x2+c′=0⇒x2ay+a′+cy+c′=0If x is rational, then the discriminant of the above equation must be a perfect square.i.e. 0−4ay+a′cy+c′ must be a perfect squarei.e. −acy2−ac′+a′cy−a′c′ must be a perfect square⇒ ac′+a′c2−4aca′c′=0             [∴  Disc =0]⇒ ac′−a′c2=0⇒ac′=a′c⇒aa′=cc′