Are the contours of the conditional bivariate normal distribution really elliptical?

Question

I understand that the contours of the bivariate normal distribution are elliptical and rotated $45$ or $135$ degrees if there is positive or negative correlation. However, I was reading the Rice textbook and I think there was an error. When stating that the bivariate normal has elliptical contours, the author wrote down the expression contained within the exponent in a simplified case with both means $0$ and variance $1$ writing: $(p^2)(x^2)-2pxy+y^2=$constant. I then saw that this is the expression contained within the exponent of the conditional bivariate normal distribution $y|X=x$ and not the joint distribution. I then checked this equation by graphing it and found that it was not an elipse. It almost was though. If the $p^2$ term is allowed to be even the slightest bit greater than its value, this equation truly is an ellipse. Can anyone here with more knowledge confirm that the conditional bivariate normal distribution does not have elliptical contours. By the way, it would seem to make sense to me that it not be elliptical since $x$ must be a constant before the graph takes the shape of a legitimate density function. But I still am confused by this.