Conic Sections Overview Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Conic Sections Overview.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The four curves—circle, ellipse, parabola, and hyperbola—obtained by slicing a double cone with a plane at different angles.

Imagine a flashlight shining on a wall. Straight on: circle. Tilted slightly: ellipse. Tilted to match the cone's edge: parabola. Tilted past the edge: hyperbola. All four shapes come from the same geometric object (a cone), just viewed from different angles.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: All four conics are unified by a single concept: they are cross-sections of a cone, classified by eccentricity. The general second-degree equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 can represent any conic.

Common stuck point: The discriminant B^2 - 4AC classifies the conic from the general equation. This is different from the quadratic discriminant b^2 - 4ac—don't confuse them.

Sense of Study hint: Compute B^2 - 4AC from the general equation. Negative means ellipse (or circle), zero means parabola, positive means hyperbola.

Worked Examples

Example 1

easy
Identify the type of conic section: \frac{x^2}{9} + \frac{y^2}{9} = 1.

Solution

  1. 1
    The equation has the form \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 with both terms positive (addition).
  2. 2
    Since a^2 = b^2 = 9, we have a = b = 3.
  3. 3
    An ellipse with a = b is a circle of radius 3 centered at the origin.

Answer

\text{Circle with radius } 3
A circle is a special case of an ellipse where both semi-axes are equal (a = b). The four conic sections — circle, ellipse, parabola, and hyperbola — are all cross-sections of a cone, distinguished by the angle at which the cutting plane intersects the cone.

Example 2

medium
Classify the conic given by 4x^2 - 9y^2 + 16x + 18y - 29 = 0.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Classify each equation: (a) x^2 + y^2 - 6x + 2y + 1 = 0, (b) x^2 - 4y + 8 = 0, (c) 9x^2 + 4y^2 = 36.

Example 2

hard
The general equation 2x^2 + 2y^2 + Bxy - 8 = 0 represents a circle only when B takes a specific value. Find that value.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

equation of circleellipsehyperbolaparabola focus directrix