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Chapter 10: Q56. (page 677)

Prove that, except for degenerate cases, the equation Ax2+Bxy+Cy2+Dx+Ey+F=0

(a) Defines a parabola if role="math" localid="1647193183886" B2-4AC=0

(b) Defines a ellipse (or circle) if role="math" localid="1647193226674" B2-4AC<0

(c) Defines a hyperbola ifB2-4AC>0

Short Answer

Expert verified

Thus, it is proved to be define a conic, except for degenerate cases.

Step by step solution

01

Step 1. Given Information

The given equation of conicAx2+Bxy+Cy2+Dx+Ey+F=0-----(1)

02

Part (a) Step 1. Proof for parabola

If AC=0,then either A=0,orC=0but not both, so equation 1may be of the form

Ax2+Bxy+Dx+Ey+F=0,A鈮�0or

Bxy+Cy2+Dx+Ey+F=0,C鈮�0

thenB2-4AC=0represent parabola.

03

Part (b) Step 1. Proof for Ellipse

If AC>0, then A and C are of the same sign. Using the results of Problem 84

in Exercise 10.3, except for the degenerate cases, the equation is an ellipse.

Thus, B2-4AC<0represent ellipse.

04

Part (c) Step 1. Proof for hyperbola

If AC<0, then A and C are of opposite sign. Using the results of Problem 86 in Exercise 10.4, except for the degenerate cases, the equation is a hyperbola.

Thus,B2-4AC>0represent hyperbola.

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