DeepConic sections

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §3.5 Parabolas, Ellipses, and Hyperbolas, #45 · Geometry · Conic sections

(a) When the plane … meets the cone …, eliminate … by…

This problem is about second-degree curves (conics).

(a) When the plane $z = a x + b y + c$ meets the cone $z^{2} = x^{2} + y^{2}$ , eliminate $z$ by squaring the plane equation. Rewrite in the form $A x^{2} + B x y + C y^{2} + D x + E y + F = 0$ .
(b) Compute $B^{2} - 4 A C$ in terms of $a$ and $b$ .
(c) Show that the plane meets the cone in an ellipse if $a^{2} + b^{2} < 1$ and a hyperbola if $a^{2} + b^{2} > 1$ (steeper).

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §3.5 Parabolas, Ellipses, and Hyperbolas, #45

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