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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. 9 (page 700)

Identify each conic without completing the square or rotating axes. $x^{2} - 6 x y + 9 y^{2} + 2 x - 3 y - 2 = 0$

Short Answer

Expert verified

The answer is Parabola.

Step by step solution

01

Step 1. Given information

The given equation of conic is $x^{2} - 6 x y + 9 y^{2} + 2 x - 3 y - 2 = 0$

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="1647880928143" $B^{2} - 4 A C = 0, P a r a b o l a B^{2} - 4 A C > 0, E l l i p s e B^{2} - 4 A C < 0, H y p e r b o l a$

On comparing the given equation with the above equation

$A = 1; B = - 6; C = 9; D = 2; E = - 3; F = - 2$

03

Step 3. Verification of conditions

Let us verify for the conic

$\begin{array}{rcl} B^{2} - 4 A C & = & {(- 6)}^{2} - 4 (1) (9) \\ = & 36 - 36 \\ = & 0 \end{array}$

Therefore, the given conic is Parabola.

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Most popular questions from this chapter

In Problems 31鈥� 42, rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation. Refer to Problems 21鈥�30 for Problems 31鈥� 40. $11 x^{2} + 10 \sqrt{3} x y + y^{2} - 4 = 0$

In Problems 43鈥�52, identify the graph of each equation without applying a rotation of axes.

2 x^{2} - 3 x y + 4 y^{2} + 2 x + 3 y - 5 = 0

Find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility.

$y^{2} = - 16 x$

True or False

To eliminate the $x y -$ term from the equation $x^{2} - 2 x y + y^{2} - 2 x + 3 y + 5 = 0$ , rotate the axes through an angle $胃$ , where $\cot 胃 = B^{2} - 4 A C$ .

To graph $y = {(x + 1)}^{2} - 4$ , shift the graph of $y = x^{2}$ to the (left/right) $_____$ unit(s) and then (up/down) $_____$ unit(s).

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