Mathematics Asked by twosigma on December 6, 2020

Consider a quadratic equation in two variables, $$f(x,y) = ax^2 + bxy + cy^2 + d$$. Let $$P$$ be a plane, so it is described by some equation $$alpha x + beta y + gamma z = delta$$ (where at least one of $$alpha, beta$$ or $$gamma$$ is nonzero).

It appears, through some examples of $$f$$ and $$P$$ that I’ve graphed, that the intersection of the graph of $$f$$ with the plane $$P$$ is a curve that looks like the graph of an ordinary one-dimensional parabola or hyperbola or general quadratic equation in one variable.

For example, the graph of $$f(x,y) = x^2 – y^2$$ seems to intersect the plane $$2x + y + 2z = 1$$ in a hyperbola, and it seems to intersect $$2x + y = 1$$ in a parabola. (In fact it seems that any vertical cross-section will produce a parabola.)

Is there a proof or counterexample of this phenomenon?

(Another (optional) question: how does this all relate to conic sections?)

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