Unformatted text preview: we have by looking at a special, familiar combination of the coeﬃcients of the
quadratic terms. We have the following theorem.
Theorem 11.11. Suppose the equation Ax2 + Bxy + Cy 2 + Dx + Ey + F = 0 describes a nondegenerate conic section.a
• If B 2 − 4AC > 0 then the graph of the equation is a hyperbola.
• If B 2 − 4AC = 0 then the graph of the equation is a parabola.
• If B 2 − 4AC < 0 then the graph of the equation is an ellipse or circle.
a Recall that this means its graph is either a circle, parabola, ellipse or hyperbola. See page 399. As you may expect, the quantity B 2 − 4AC mentioned in Theorem 11.11 is called the discriminant
of the conic section. While we will not attempt to explain the deep Mathematics which produces this
‘coincidence’, we will at least work through the proof of Theorem 11.11 mechanically to show that it
is true.5 First note that if the coeﬃcient B = 0 in the equation Ax2 + Bxy + Cy 2 + Dx + Ey + F = 0,
Theorem 11.11 reduces to the result presented in Exercise 10 in Sec...
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