Conic Sections Overview Math Example 3

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Example 3

medium
Classify each equation: (a) x2+y2−6x+2y+1=0x^2 + y^2 - 6x + 2y + 1 = 0, (b) x2−4y+8=0x^2 - 4y + 8 = 0, (c) 9x2+4y2=369x^2 + 4y^2 = 36.

Solution

  1. 1
    (a) A=1A = 1, C=1C = 1, A=CA = C with no xyxy term: circle. (b) Only x2x^2 term, no y2y^2: parabola. (c) Both positive with A=9≠C=4A = 9 \neq C = 4: ellipse.
  2. 2
    Verify: (a) Circle, (b) Parabola (opens upward), (c) Ellipse x24+y29=1\frac{x^2}{4} + \frac{y^2}{9} = 1.

Answer

(a) Circle, (b) Parabola, (c) Ellipse\text{(a) Circle, (b) Parabola, (c) Ellipse}
Quick classification rules: same positive coefficients on x2x^2 and y2y^2 means circle; different positive coefficients means ellipse; one squared term missing means parabola; opposite signs on squared terms means hyperbola.

About Conic Sections Overview

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

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More Conic Sections Overview Examples

Example 1 easy

Identify the type of conic section: [formula].

Example 2 medium

Classify the conic given by [formula].

Example 4 hard

The general equation [formula] represents a circle only when [formula] takes a specific value. Find

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