Key takeaway: Conic sections are the four curves a flat plane carves from a double cone at different tilts: circle, ellipse, parabola and hyperbola.
Grade 12 – PrecalculusG-GPE.A.3Geometry and Measurement
Definition of Conic Sections
Conic sections get their name from a simple experiment: slice a double cone with a flat plane. A level cut gives a circle. Tilt the plane a little and the cut stretches into an ellipse. Tilt it until it is parallel to the cone’s slanted side and a parabola appears. Tilt further so the plane cuts both halves and you get the two branches of a hyperbola.
All four are described by a second-degree equation in \(x\) and \(y\). Their squared terms tell you which curve you have: matching positive coefficients suggest a circle, unequal positive ones an ellipse, opposite signs a hyperbola, and only one squared variable a parabola.
Formula and notation
Two of the four standard forms centered at the origin:
\[\text{circle: } x^2+y^2=r^2,\qquad \text{parabola: } y=ax^2.\]
An ellipse and a hyperbola both have two fractions on the left, with \(x^2\) over \(a^2\) and \(y^2\) over \(b^2\), and 1 on the right. A plus between the fractions gives the ellipse; a minus gives the hyperbola.
Key facts
- Among conic sections, the circle is a special ellipse with equal axes.
- Only the hyperbola has two separate branches and asymptotes.
- Orbits of planets are ellipses; the path of a thrown ball is part of a parabola.
- Each curve can be defined by distances to special points called foci.
Conic Sections example
Classify each equation: (a) \(9x^2+4y^2=36\), (b) \(y=2x^2-8x+3\), (c) \(5x^2-3y^2=45\).
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(a) Both squares are positive with different coefficients. Dividing everything by 36 leaves denominators 4 and 9, so the graph is an ellipse taller than it is wide.
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(b) Only \(x\) is squared, so the graph is a parabola opening upward.
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(c) The squares have opposite signs. Dividing by 45 leaves denominators 9 and 15 with a minus between them: a hyperbola opening left and right.
(a) ellipse, (b) parabola, (c) hyperbola.
Common mistake
Watch out
Wyatt sees \(9x^2+4y^2=36\) and calls it a circle, because both variables are squared and added.
Why it is wrong
A circle needs equal coefficients on the squared terms. Here the coefficients 9 and 4 differ, so the \(y\)-radius is 3 and the \(x\)-radius is 2, an ellipse. For all conic sections of this type, divide so the right side is 1 and compare the denominators.
Related terms and lessons
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