Conic Sections Overview Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Classify the conic given by 4x2βˆ’9y2+16x+18yβˆ’29=04x^2 - 9y^2 + 16x + 18y - 29 = 0.

Solution

  1. 1
    Look at the x2x^2 and y2y^2 coefficients: A=4A = 4 and C=βˆ’9C = -9. They have opposite signs.
  2. 2
    When AA and CC have opposite signs, the conic is a hyperbola.
  3. 3
    Verify by computing the discriminant B2βˆ’4ACB^2 - 4AC where B=0B = 0: 0βˆ’4(4)(βˆ’9)=144>00 - 4(4)(-9) = 144 > 0, confirming a hyperbola.

Answer

Hyperbola\text{Hyperbola}
For the general second-degree equation Ax2+Bxy+Cy2+Dx+Ey+F=0Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0: if B2βˆ’4AC<0B^2 - 4AC < 0 and A=CA = C, it is a circle; if B2βˆ’4AC<0B^2 - 4AC < 0 and Aβ‰ CA \neq C, it is an ellipse; if B2βˆ’4AC=0B^2 - 4AC = 0, it is a parabola; if B2βˆ’4AC>0B^2 - 4AC > 0, it is a hyperbola.

About Conic Sections Overview

The four curvesβ€”circle, ellipse, parabola, and hyperbolaβ€”obtained by slicing a double cone with a plane at different angles.

Learn more about Conic Sections Overview β†’

More Conic Sections Overview Examples

Example 1 easy

Identify the type of conic section: [formula].

Example 3 medium

Classify each equation: (a) [formula], (b) [formula], (c) [formula].

Example 4 hard

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

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