*This preview shows
page 1. Sign up
to
view the full content.*

**Unformatted text preview: **as the opposite sign of A C . The result
now follows by applying Exercise 10 in Section 7.5.
Example 11.6.3. Use Theorem 11.11 to classify the graphs of the following non-degenerate conics.
√
1. 21x2 + 10xy 3 + 31y 2 = 144
√
√
2. 5x2 + 26xy + 5y 2 − 16x 2 + 16y 2 − 104 = 0
3. 16x2 + 24xy + 9y 2 + 15x − 20y = 0
Solution. This is a straightforward application of Theorem 11.11. 834 Applications of Trigonometry √
√
1. We have A = 21, B = 10 3 and C = 31 so B 2 − 4AC = (10 3)2 − 4(21)(31) = −2304 < 0.
Theorem 11.11 predicts the graph is an ellipse, which checks with our work from Example
11.6.1 number 2.
2. Here, A = 5, B = 26 and C = 5, so B 2 − 4AC = 262 − 4(5)(5) = 576 > 0. Theorem 11.11
classiﬁes the graph as a hyperbola, which matches our answer to Example 11.6.2 number 1.
3. Finally, we have A = 16, B = 24 and C = 9 which gives 242 − 4(16)(9) = 0. Theorem 11.11
tells us that the graph is a parabola, matching our result from Example 11.6.2 number 2. 11.6.2 The Polar Form of Conics...

View
Full
Document