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Learn Conic Sections

Circles, ellipses, parabolas and hyperbolas from one sliced cone.

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Conic sections are the curves you get by slicing a double cone with a plane: circles, ellipses, parabolas and hyperbolas. Each has a geometric definition, such as an ellipse being the points whose distances to two foci add to a constant, and a standard equation that reveals its center, size and shape. They describe planetary orbits, satellite dishes, headlight reflectors and the paths of thrown objects. Most of the work is completing the square to turn a general equation into standard form, then reading off and graphing the key features. Learnr builds a conic sections course at your level and teaches each curve in short interactive lessons with graphs.

How to learn Conic Sections, step by step

Review the distance formula and completing the square. Every conic definition is about distances, and nearly every standard form comes from completing the square. Make both automatic before starting the curves.
Start with circles. Rewrite x squared + y squared - 6x + 4y - 12 = 0 by completing the square in x and in y to get (x - 3) squared + (y + 2) squared = 25: center (3, -2), radius 5.
Learn parabolas from focus and directrix. In y squared = 4px, the focus is at (p, 0) and the directrix is x = -p. Build one from string-and-ruler reasoning to see why dishes concentrate signals at the focus.
Separate ellipses from hyperbolas carefully. An ellipse adds its squared terms and uses c squared = a squared minus b squared for the foci. A hyperbola subtracts them and uses c squared = a squared plus b squared.
Graph hyperbolas with the box method. Draw a rectangle using a and b, run the asymptotes through its corners, then sketch the branches from the vertices. The box makes accurate sketches quick.
Identify conics from mixed equations daily. Practice telling the four apart from general equations by the signs and coefficients of the squared terms. A short daily Learnr set, kept up with a streak, makes this identification instant.

What you'll learn

Identify a conic from its equation
Rewrite equations in standard form by completing the square
Graph circles and ellipses from center and radii
Find vertex, focus and directrix of parabolas
Graph hyperbolas with their asymptotes and foci

What's inside your Conic Sections course

A sample path Learnr builds for Conic Sections. Your real course adapts to your goal and level.

1. Slicing a cone and the distance formula
2. Circles
3. Parabolas: focus and directrix
4. Ellipses
5. Hyperbolas
6. Identifying conics and real applications

How long does it take to learn Conic Sections?

Conic sections usually take about 4 to 6 weeks at 3 to 4 hours a week as part of an Algebra 2 or precalculus course. Learners strong in completing the square can move faster.

What you need before you start

Graphing in the coordinate plane, the distance formula, completing the square and solving quadratic equations. Familiarity with parabolas from algebra helps.

Common mistakes to avoid

Mixing up the focal formulas: ellipses use c squared = a squared minus b squared, hyperbolas use c squared = a squared plus b squared.
Reading the radius as r squared, so a circle with right side 25 gets radius 25 instead of 5.
Getting the center's signs backward. (x - 3) squared means the center's x-coordinate is +3, and (y + 2) squared means it is -2.

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Conic Sections questions, answered

Where are conic sections used in real life?

Planets orbit in ellipses, satellite dishes and flashlight reflectors use parabolas, and some navigation systems and cooling tower shapes involve hyperbolas.

How can I tell which conic an equation is?

For Ax squared + Cy squared + Dx + Ey + F = 0 with no xy term: only one squared term means a parabola, equal A and C means a circle, same signs means an ellipse and opposite signs mean a hyperbola.