Number of solutions for a system sounds odd

Question

I had a system of 2 degree equations today in exam, two equations, with 2 variables. I had solved for "x" first, then "y", I cannot remember the system (sorry), however, I got 2 solutions for "x", and this is what i got:

x = 5, or x = -3

by putting "x" in a one of the two equations:

when x = 5, y = 4 or -4, so, the first two solutions were:

{(5,4) , (5,-4)}

now when I've put -3, I got "y" as zero, so i got a one solution because zero isn't either positive or negative; so i got this one extra solution: (-3, 0),

now all I got is 3 solutions in a system of non-linear equations, is this normal?

Answer 1

The graph of a 2nd-degree equation will be a parabola, circle, ellipse or hyperbola (or some degenerate version.) We need to imagine a situation where two of these curves could intersect in exactly 3 points. For instance takes $y=x^2$, a parabola with vertex at the origin and arms pointing up, and $x^2+(y-1)^2=1$, a circle of radius $1$ centered at $(0,1)$. Then the arms of the parabola cut the circle at $(-1,1)$ and $(1,1)$. The 3rd point of intersection is at $(0,0)$ where the two curves are tangent.

It's sort of rare for two, randomly chosen curves to be tangent, so I'm not sure this counts as "normal". But it's certainly possible.