Q 29. In Problem, determine the approp... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 10: Q 29. (page 677)

In Problem, determine the appropriate rotation formulas to use so that the new equation contains no $x y -$ term.
$25 x^{2} - 36 x y + 40 y^{2} - 12 \sqrt{13} x - 8 \sqrt{13} y = 0$

Short Answer

Expert verified

$\begin{array}{rcl} x & = & \frac{\sqrt{13}}{13} (3 x^{'} - 2 y^{'}) \\ y & = & \frac{\sqrt{13}}{13} (2 x^{'} + 3 y^{'}) \end{array}$

Step by step solution

Step 1. Given information

An equation is $25 x^{2} - 36 x y + 40 y^{2} - 12 \sqrt{13} x - 8 \sqrt{13} y = 0$ .

Step 2. Determine the appropriate rotation formulas.

The appropriate acute angle $胃$ through which to rotate the axes, so that the $x y$ term is removed is $\cot (2 胃) = \frac{A - C}{B}$ .

Substitute $A = 25, B = - 36, C = 40$ in this equation to get $胃$ .

$\begin{array}{rcl} \cot (2 胃) & = & \frac{25 - 40}{- 36} \\ = & \frac{5}{12} \\ \cot (2 胃) & = & \frac{5}{12} \\ \sin (2 胃) & = & \frac{12}{13} \\ \cos (2 胃) & = & \frac{5}{13} \end{array}$

$\begin{array}{rcl} \cos^{2} 胃 & = & \frac{1 + (\frac{5}{13})}{2} \\ = & \frac{9}{13} \\ \cos 胃 & = & \frac{3}{\sqrt{13}} \end{array}$

$\begin{array}{rcl} \sin^{2} 胃 & = & \frac{1 - (\frac{5}{13})}{2} \\ = & \frac{4}{13} \\ \sin 胃 & = & \frac{2}{\sqrt{13}} \end{array}$

$\begin{array}{rcl} x & = & x^{'} + \frac{3}{\sqrt{13}} - y^{'} + \frac{2}{\sqrt{13}} \\ = & \frac{\sqrt{13}}{13} (3 x^{'} - 2 y^{'}) \\ y & = & x^{'} 路 \frac{2}{\sqrt{13}} + y^{'} 路 \frac{3}{\sqrt{13}} \\ = & \frac{\sqrt{13}}{13} (2 x^{'} + 3 y^{'}) \end{array}$

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91影视

In Problem, determine the appropriate rotation formulas to use so that the new equation contains no $x y -$ term.
$25 x^{2} - 36 x y + 40 y^{2} - 12 \sqrt{13} x - 8 \sqrt{13} y = 0$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Determine the appropriate rotation formulas.

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91影视

In Problem, determine the appropriate rotation formulas to use so that the new equation contains no xy-term.25x2-36xy+40y2-1213x-813y=0

Short Answer

Step by step solution

Step 1. Given information

Step 2. Determine the appropriate rotation formulas.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Applied Mathematics

Discrete Mathematics

Calculus

Pure Maths

Geometry

Study anywhere. Anytime. Across all devices.

In Problem, determine the appropriate rotation formulas to use so that the new equation contains no $x y -$ term.
$25 x^{2} - 36 x y + 40 y^{2} - 12 \sqrt{13} x - 8 \sqrt{13} y = 0$