### A note on points of intersection

An interesting fact I thought about today. Let us suppose we have to determine the points of intersection of the cartesian equations and . What we generally do is .

Why? Because the points of intersection will, on substitution, make both sides equal. On substitution, the values obtained on both sides will be .

But does the equation only determine the points of intersection of the figures and ? No. It also determines the points of intersection of the figures and for every .

Shocking, isn’t it? Determining only the points of intersection relevant to can be done by substitution of the points obtained in the two equations separately.