Practice Conic Sections Overview in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick 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.

Example 1

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

Example 2

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

Example 3

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 4

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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