Introduction to Conic Sections
What You’ll Learn
Section titled “What You’ll Learn”In this lesson you’ll learn what conic sections are, how they are formed, and get an overview of the four main types: circles, parabolas, ellipses, and hyperbolas.
The Concept
Section titled “The Concept”Conic sections are curves formed by intersecting a plane with a double cone (two cones joined at their tips). The shape depends on the angle of the cut.
Each double cone above is shown from the side. The colored line represents the cutting plane. A horizontal cut (blue) gives a circle. An angled cut (green) gives an ellipse. A cut parallel to the cone’s side (orange) gives a parabola. A nearly vertical cut through both halves (purple) gives a hyperbola.
The diagram shows the four conic sections. A circle (blue) is perfectly round, equal width in all directions. An ellipse (green) is a stretched circle, wider in one direction. A parabola (orange) is an open U-shape that extends to infinity. A hyperbola (purple) has two separate branches that curve away from each other.
How each is formed by slicing a cone:
- Circle: plane cuts parallel to the base of the cone.
- Ellipse: plane cuts at an angle, but doesn’t go parallel to the side.
- Parabola: plane cuts parallel to the side of the cone.
- Hyperbola: plane cuts through both halves of the double cone.
Each conic section can also be defined algebraically as a second-degree equation:
In Algebra 2, we usually study the non-rotated cases (where B = 0).
Worked Example
Section titled “Worked Example”Identify the conic section from its equation:
1. x² + y² = 25
Both x² and y² have the same positive coefficient (1) → Circle with radius 5.
2. y = x² − 4x + 3
Only one variable is squared (x²) → Parabola.
3. x²/9 + y²/4 = 1
Both x² and y² are positive but have different denominators → Ellipse.
4. x²/16 − y²/9 = 1
x² and y² have opposite signs (one positive, one negative) → Hyperbola.
The four graphs above show the actual equations from the worked examples. The blue circle is perfectly round with radius 5. The orange parabola opens upward with vertex at (2, −1). The green ellipse is wider than it is tall (a = 3, b = 2). The purple hyperbola has two branches opening left and right.
Quick identification guide:
| What you see | Conic section |
|---|---|
| x² + y² (same coefficient) | Circle |
| Only x² or only y² | Parabola |
| x² + y² (different coefficients) | Ellipse |
| x² − y² (opposite signs) | Hyperbola |
Real-World Application
Section titled “Real-World Application”Conic sections appear in many real-world situations:
- Parabolas: satellite dishes, headlights, suspension bridges, projectile paths.
- Ellipses: planetary orbits, whispering galleries, certain stadium designs.
- Circles: wheels, clocks, round tables, manhole covers.
- Hyperbolas: navigation systems (LORAN), cooling towers, some telescope designs.
Example: a satellite dish is shaped like a parabola because it reflects incoming signals to a single focal point.