Degenerate Conic Sections

High School

Definition

Made up of a point, a line, and intersecting lines. Degenerate conic sections refer to plane figures that you get from the intersection of a double cone that has a plane passing through its apex.

These sections can be expressed in an equation:

\( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \)

Worked examples

\(x^2 + y^2 = 0\)

This simplifies to \(x = 0\) and \(y = 0\), a single point at the origin — a degenerate circle.

\(x^2 - y^2 = 0\)

Factoring gives \((x - y)(x + y) = 0\), so \(y = x\) and \(y = -x\) — two intersecting lines.

\(x^2 - 4 = 0\)

This factors to \((x - 2)(x + 2) = 0\), giving \(x = 2\) and \(x = -2\) — two parallel lines.

Common mistakes

All conic equations with \(Ax^2 + Cy^2\) are non-degenerate → When the right-hand side is zero, the conic may be degenerate Check if the equation simplifies to a point, line, or intersecting lines.
\(x^2 + y^2 = 0\) has no solution → \(x^2 + y^2 = 0\) has exactly one solution: \((0, 0)\) Zero squared plus zero squared equals zero, giving the origin as a degenerate point.
Two parallel lines \(x = \pm 2\) form a hyperbola → Two parallel lines are a degenerate case, not a standard hyperbola Standard hyperbolas have two separate branches; parallel lines are the limiting degenerate form.

Where you'll use it next

Degenerate conics appear when classifying all solutions to quadratic equations in two variables, in limits and tangency problems in calculus, and when analyzing edge cases in algebraic geometry.

Found in 1 StudyPug lesson

Conics - circle

Grade 12 Math

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