Conic Sections
In short: Conic sections are the four curves, circle, ellipse, parabola and hyperbola, made by slicing a double cone with a plane at different angles, each with its own type of equation.
PrecalculusGeometryG-GPE.A.3
What is Conic Sections?
The conic sections get their name from a cone. Slice a double cone straight across and you get a circle; tilt the cut slightly for an ellipse; cut parallel to the side for a parabola; and cut steeply through both halves for a hyperbola, which has two branches.
Each one fits an equation of the type \(Ax^2 + Cy^2 + Dx + Ey + F = 0\), and the signs of \(A\) and \(C\) reveal which curve you have.
How to spot it
- An equation of conic sections contains x² and y² (or only one of them) with no higher powers and no xy product.
- A question says to identify the curve and gives only its equation.
- A real setting involves satellite dishes, planetary orbits, bridge arches or cooling towers.
- Completing the square is needed to read the center and size of a curve.
The method
- Sort the \(x\) terms and \(y\) terms, then study the signs on \(x^2\) and \(y^2\).
- With a single squared variable you are looking at a parabola; with two, compare the coefficients.
- Same sign and size means circle, same sign but different size means ellipse, and opposite signs mean hyperbola.
- Complete the square for each conic to reach standard form and read off the center.
Conic Sections: solved example
Problem
Identify each curve: (a) \(x^2 + y^2 – 6x = 7\); (b) \(9x^2 – 4y^2 = 36\); (c) \(y = 2x^2 – 8x + 1\).
Solution
(a) Complete the square: \((x – 3)^2 + y^2 = 16\), a circle with center \((3, 0)\) and radius 4. (b) Divide by 36: \(\frac{x^2}{4} – \frac{y^2}{9} = 1\), where the minus sign marks a hyperbola.
(c) Only \(x\) is squared, so it is a parabola. The answers are circle, hyperbola, parabola.
Try it
Question 1
What curve is \(5x^2 + 5y^2 = 45\)?
Show the answer
Divide by 5: \(x^2 + y^2 = 9\), a circle of radius 3.
Question 2
What curve is \(16x^2 + 25y^2 = 400\)?
Show the answer
Divide by 400: \(\frac{x^2}{25} + \frac{y^2}{16} = 1\), an ellipse with semi-axes 5 and 4.
Conic Sections vs. Parabola
A parabola is a single member of this family, a curve with one branch and one squared variable. The family also holds closed curves, the circle and ellipse, and the two-branch hyperbola. A quadratic function gives only the parabola, while the general second-degree equation covers all four.
Keep going
Lessons
Worksheets and quizzes
- Conic Sections – Worksheet
- Conic Sections – Quiz
- Equations of Circles and Parabolas – Worksheet
- Equations of Circles and Parabolas – Quiz
Related words
Whole dictionary: A–Z · Precalculus · Geometry