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Intersection of a quadratic and a plane is a quadratic?

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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