A note on points of intersection

by ayushkhaitan3437

An interesting fact I thought about today. Let us suppose we have to determine the points of intersection of the cartesian equations f(x,y)=c and g(x,y)=d. What we generally do is f(x,y)-c=g(x,y)-d.

Why? Because the points of intersection (a,b) will, on substitution, make both sides equal. On substitution, the values obtained on both sides will be 0.

But does the equation f(x,y)-c=g(x,y)-d only determine the points of intersection of the figures f(x,y)=c and g(x,y)=d? No. It also determines the points of intersection of the figures f(x,y)=x+r and g(x,y)=d+r for every r\in\Bbb{R}.

Shocking, isn’t it? Determining only the points of intersection relevant to r=0 can be done by substitution of the points obtained in the two equations separately.

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