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Home Limits, Analytic Geometry, and Approximations Second Degree Curves Examples Example: Discriminant Test (Applied on a Hyperbola)
 
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Example: Discriminant Test (Applied on a Hyperbola)

Sketch the curve

4x2 - y2 - 16x - 2y + 11 = 0.

Step 1

Apply the Discriminant Test to determine the type of curve.

B2 - 4AC = 02 - 4 · 4 · (-1) = 16.

The discriminant is positive, so the graph is a hyperbola.

Step 2

Simplify by completing the squares. This is done by putting

05_limits_g_approx-345.gif

and writing the original equation in terms of X and Y.

05_limits_g_approx-346.gif

4(X + 2)2 - (Y - 1)2 - 16(X + 2) - 2(Y - 1) + 11 = 0

4(X2 + 4X + 4) - 16(X + 2) - (Y2 - 2Y + 1) - 2(Y - 1) + 11 = 0.

The X and Y terms cancel, and

4X2 + 16 - 32 - Y2 - 1 + 2 + 11 = 0,

4X2 - Y2 - 4 = 0.
Step 3 Draw dotted lines for the X and Y axes, and sketch the curve as in Section 5.5. This is a hyperbola in the (X, Y)-plane. The X-axis is the line Y = 0, or Y = -1. The Y-axis is the line X = 0, or x = 2. The graph is shown in Figure 5.6.2.

05_limits_g_approx-347.gif

Figure 5.6.2: Example1: 4x2 - y2 - 16x - 2y + 11 = 0
Home Limits, Analytic Geometry, and Approximations Second Degree Curves Examples Example: Discriminant Test (Applied on a Hyperbola)

Last Update: 2006-11-15