Chapter 10: Q 27. (page 677)

In Problem, determine the appropriate rotation formulas to use so that the new equation contains no $x y -$ term.
$4 x^{2} - 4 x y + y^{2} - 8 \sqrt{5} x - 16 \sqrt{5} y = 0$

Short Answer

Expert verified

$x = \frac{\sqrt{5}}{5} (x^{'} - 2 y^{'}) y = \frac{\sqrt{5}}{5} (2 x^{'} + y^{'})$

Step by step solution

Step 1. Given information

An equation is $4 x^{2} - 4 x y + y^{2} - 8 \sqrt{5} x - 16 \sqrt{5} y = 0$ .

Step 2. Determine the appropriate rotation formulas.

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

Substitute $A = 4, B = - 4, C = 1$ in this equation to get $θ$ .

$\begin{array}{rcl} \cot (2 θ) & = & \frac{4 - 1}{- 4} \\ = & - \frac{3}{4} \end{array}$

$\begin{array}{rcl} \cos^{2} θ & = & \frac{1 + (- \frac{3}{5})}{2} \\ = & \frac{1}{5} \\ \cos^{2} θ & = & \frac{1}{5} \\ \sin^{2} θ & = & \frac{1 - (- \frac{3}{5})}{2} \\ = & \frac{4}{5} \\ \sqrt{\sin^{2} θ} & = & \sqrt{\frac{4}{5}} \\ \sin θ & = & \frac{2}{\sqrt{5}} \\ x & = & x^{'} \cdot \frac{1}{\sqrt{5}} - y^{'} \cdot \frac{2}{\sqrt{5}} \\ = & \frac{\sqrt{5}}{5} (x^{'} - 2 y^{'}) \\ y & = & x^{'} \cdot \frac{2}{\sqrt{5}} + y^{'} \cdot \frac{1}{\sqrt{5}} \\ = & \frac{\sqrt{5}}{5} (2 x^{'} + y^{'}) \end{array}$

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In Problem, determine the appropriate rotation formulas to use so that the new equation contains no $x y -$ term.
$4 x^{2} - 4 x y + y^{2} - 8 \sqrt{5} x - 16 \sqrt{5} y = 0$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Determine the appropriate rotation formulas.

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In Problem, determine the appropriate rotation formulas to use so that the new equation contains no xy-term.4x2-4xy+y2-85x-165y=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

Logic and Functions

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Probability and Statistics

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Calculus

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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.
$4 x^{2} - 4 x y + y^{2} - 8 \sqrt{5} x - 16 \sqrt{5} y = 0$