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Pre-Calculus Conic sections

Conic sections from the double-napped cone (geometric definition)

20 practice questions 0 video lessons Theory + worked examples

Conic Sections from the Cone

Common Core Pre-Calculus • Standard G-GPE • Conic Sections

Conic Sections from the Cone is the opening topic of Conic Sections in the Common Core State Standards. It is aligned to Standard G-GPE, which requires students to understand the conic sections as slices of a cone.

The conic sections — circle, ellipse, parabola, and hyperbola — are the curves formed when a plane slices a double-napped cone.

Common Core Pre-Calculus › Conic Sections › Conic Sections from the Cone  —  Standard G-GPE

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Theory

The conic sections are the curves formed when a plane slices a double-napped cone. The angle of the cut decides the curve:

  • Circle: a horizontal slice, perpendicular to the axis.
  • Ellipse: a tilted slice through one nap.
  • Parabola: a slice parallel to the cone's slant side.
  • Hyperbola: a steep slice cutting both naps — giving two branches.
One family, four shapes. All four conics come from the same cone; the tilt of the plane is the only thing that changes.
Conic sections from a double cone Slicing a double-napped cone at different angles produces a circle, ellipse, parabola, or hyperbola. slice
A plane slicing the cone produces a conic section.
The four conics The four conics The four conics horizontal slice → circle tilted slice → ellipse parallel to side → parabola steep (both naps) → hyperbola
Which conic each slice gives.

Every conic satisfies a second-degree equation:

\[Ax^2+Bxy+Cy^2+Dx+Ey+F=0\]
every conic is a second-degree equation in x and y
The type is set by \(A\) and \(C\) (with \(B=0\) for un-rotated conics) — explored in the general-form topic.

How to recognize a conic from its slice

  1. Perpendicular to the axis \(\to\) circle.
  2. Tilted through one nap \(\to\) ellipse.
  3. Parallel to the slant side \(\to\) parabola.
  4. Through both naps \(\to\) hyperbola.
Example 1 — Identify by the slice
What conic results from a plane parallel to the cone's slant side?
Solution

A cut parallel to the side yields a single open curve.

\text{parallel to side}\(\to\)\text{parabola}
a slice parallel to the side gives a parabola
Example 2 — Circle vs ellipse
How does the slice differ for a circle versus an ellipse?
Solution

A circle needs a slice perpendicular to the axis; tilting the plane stretches it into an ellipse.

\text{perpendicular}\(\to\)\text{circle}
\text{tilted}\(\to\)\text{ellipse}
perpendicular slice gives a circle, tilted gives an ellipse
Example 3 — Hyperbola
Why does a hyperbola have two branches?
Solution

A steep enough plane cuts both naps of the double cone, producing two separate curves.

\text{both naps}\(\to\)\text{two branches}
cutting both naps of the cone gives two branches
Example 4 — A degenerate case
What if the slicing plane passes through the cone's vertex?
Solution

You get a degenerate conic: a single point, a line, or a pair of intersecting lines.

\text{through vertex}\(\to\)\text{point / line(s)}
a plane through the vertex gives a degenerate conic

Common pitfalls

A circle is a special ellipse. It needs a perfectly perpendicular slice.
Hyperbolas need both naps. That is why they have two separate branches.
Through the vertex gives a degenerate conic — a point or line(s), not a full curve.

Frequently asked questions

What are the conic sections?

The circle, ellipse, parabola, and hyperbola — the curves formed by slicing a double-napped cone with a plane.

What determines which conic you get?

The angle of the slicing plane relative to the cone's axis and slant side.

Why does a hyperbola have two branches?

Because a steep plane cuts both napes of the double cone, producing two separate curves.

What is a degenerate conic?

The point, line, or pair of lines you get when the plane passes through the cone's vertex.

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