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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 10: Q. 8 (page 700)

Identify each conic without completing the square or rotating axes. ${3 x}^{2} - x y + 2 y^{2} + 3 y + 1 = 0$

Short Answer

Expert verified

The answer is Ellipse.

Step by step solution

01

Step 1. Given information

The given equation of conic is ${3 x}^{2} - x y + 2 y^{2} + 3 y + 1 = 0$

02

Step 2. Identify the conic without completing the square or rotating axes.

If the equation is in the form of $A x^{2} + B x y + C y^{2} + D x + E y + F = 0$

Defines,

role="math" localid="1647880942942" $B^{2} - 4 A C = 0, P a r a b o l a B^{2} - 4 A C > 0, E l l i p s e B^{2} - 4 A C < 0, H y p e r b o l a$

On comparing the given equation with the above equation

$A = 3; B = - 1; C = 2; D = 0; E = 3; F = 1$

03

Step 3. Verification of conditions

Let us verify for the conic

$\begin{array}{rcl} B^{2} - 4 A C & = & {(- 1)}^{2} - 4 (3) (2) \\ = & 1 - 24 \\ = & - 23 < 0 \end{array}$

Therefore, the given conic is Ellipse.

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Most popular questions from this chapter

In Problem, identify the graph of each equation without completing the squares.

$6 x^{2} + 3 y^{2} - 12 x + 6 y = 0$

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True or False

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