Concyclic iff Opposite Slopes

Let $E$ be a non-circular conic whose axes of symmetry are parallel or perpendicular to the coordinate axes, and let $A,B,C,D$ be four distinct points on $E$ such that the lines $AC$ and $BD$ meet and both have slopes. Prove that $A,B,C,D$ are concyclic if and only if the slopes of $AC$ and $BD$ are negatives of each other.