Chapter 10: Q.23 (page 699)

In Problems 19 –23, identify each conic without completing the squares and without applying a rotation of axes.
$x^{2} - 2 x y + 3 y^{2} + 2 x + 4 y - 1 = 0$

Short Answer

Expert verified

The conic represented by the given equation is an ellipse.

Step by step solution

Step 1. Given Information

Given to identify the conic without completing the squares and without applying a rotation of axes.

$x^{2} - 2 x y + 3 y^{2} + 2 x + 4 y - 1 = 0$

Step 2. Calculation

An equation of the form $A x^{2} + B x y + C y^{2} + D x + E y + F = 0$ defines:

a parabola if $B^{2} - 4 A C = 0$
an ellipse/ a circle if $B^{2} - 4 A C < 0$
a hyperbola if $B^{2} - 4 A C > 0$

In the given equation, $A = 1, B = - 2, C = 3$

Plugging the values,

$\begin{array}{rcl} B^{2} - 4 A C & = & {(- 2)}^{2} - 4 (1) (3) \\ = & 4 - 12 \\ = & - 8 \end{array}$

Since $B^{2} - 4 A C < 0$ , the given conic is an ellipse or a circle.

Since $A \neq C$ , the conic is an ellipse.

Step 3. Conclusion

Hence, the conic represented by the given equation is an ellipse.

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In Problems 19 –23, identify each conic without completing the squares and without applying a rotation of axes.
$x^{2} - 2 x y + 3 y^{2} + 2 x + 4 y - 1 = 0$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

Step 3. Conclusion

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In Problems 19 –23, identify each conic without completing the squares and without applying a rotation of axes.x2-2xy+3y2+2x+4y-1=0

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

Step 3. Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Probability and Statistics

Statistics

Discrete Mathematics

Applied Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

In Problems 19 –23, identify each conic without completing the squares and without applying a rotation of axes.
$x^{2} - 2 x y + 3 y^{2} + 2 x + 4 y - 1 = 0$