Number of common chords of the two parabolas

Question

How can we find common chords to the parabolas $$ (y-2)=(x-3)^2$$ and $$(x-2)=(y-3)^2$$ without drawing graphs.

What i have done is i have subtracted both of them and i got

$$(y-x)=(x-y)(x+y-6)$$ $\implies$

$$(x-y)(x+y-5)=0$$ Hence

$x=y$ and $x+y=5$ are the two common chords. How can find whether there are any other chords

Answer 1

Hint.

Find the point of intersection of the two line with one parabola. You find $4$ points $A,B,C,D$. All the combinations $AB$, $AC$, $AD$, $BC$, $BD$, $CD$ are common chords. And there are no other since these points are the only common points of the two parabolas.