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Q33.

Expert-verified
Found in: Page 677

### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

# In Problems 31– 42, rotate the axes so that the new equation contains no $xy$-term. Analyze and graph the new equation. Refer to Problems 21–30 for Problems 31– 40.$5{x}^{2}+6xy+5{y}^{2}-8=0$

$x=\frac{1}{\sqrt{2}}x\text{'}-\frac{1}{\sqrt{2}}y\text{'}\phantom{\rule{0ex}{0ex}}y=\frac{1}{\sqrt{2}}x\text{'}+\frac{1}{\sqrt{2}}y\text{'}$

See the step by step solution

## Step 1. Given Information

The given equation is $5{x}^{2}+6xy+5{y}^{2}-8=0$.

We need to determine the appropriate rotation formula so that new equation does not contain $xy$ term.

## Step 2. Finding acute angle

Here $A=5,B=6,C=5$

$cot2\theta =\frac{5-5}{6}=0$

Where

$cot2\theta =0\phantom{\rule{0ex}{0ex}}2\theta =90\phantom{\rule{0ex}{0ex}}\theta ={45}^{\circ }$

## Step 3. Finding the values of x and y

$x=x\text{'}\mathrm{cos}{45}^{\circ }-y\text{'}\mathrm{sin}{45}^{\circ }\phantom{\rule{0ex}{0ex}}x=\frac{1}{\sqrt{2}}x\text{'}-\frac{1}{\sqrt{2}}y\text{'}$

and

$y=x\text{'}\mathrm{sin}{45}^{\circ }+y\text{'}\mathrm{cos}{45}^{\circ }\phantom{\rule{0ex}{0ex}}y=\frac{1}{\sqrt{2}}x\text{'}+\frac{1}{\sqrt{2}}y\text{'}$

## Step 4. Substituting these values into the original equation

$5{x}^{2}+6xy+5{y}^{2}-8=0$

$\frac{5}{2}{\left(x\text{'}-y\text{'}\right)}^{2}+6×\frac{1}{\sqrt{2}}\left(x\text{'}-y\text{'}\right)×\frac{1}{\sqrt{2}}\left(x\text{'}+y\text{'}\right)+\frac{5}{2}{\left(x\text{'}-y\text{'}\right)}^{2}-8=0\phantom{\rule{0ex}{0ex}}\frac{5}{2}\left(x{\text{'}}^{2}-2x\text{'}y\text{'}+y{\text{'}}^{2}\right)+3\left(x{\text{'}}^{2}-y{\text{'}}^{2}\right)+\frac{5}{2}\left(x{\text{'}}^{2}+2x\text{'}y\text{'}+y{\text{'}}^{2}\right)=8$

## Step 5. Simplify the above Equation

$\frac{5}{2}\left(x{\text{'}}^{2}-2x\text{'}y\text{'}+y{\text{'}}^{2}\right)+3\left(x{\text{'}}^{2}-y{\text{'}}^{2}\right)+\frac{5}{2}\left(x{\text{'}}^{2}+2x\text{'}y\text{'}+y{\text{'}}^{2}\right)=8\phantom{\rule{0ex}{0ex}}\frac{5}{2}x{\text{'}}^{2}-5x\text{'}y\text{'}+\frac{5}{2}y{\text{'}}^{2}+3x{\text{'}}^{2}-3y{\text{'}}^{2}+\frac{5}{2}x{\text{'}}^{2}+5x\text{'}y\text{'}+\frac{5}{2}y{\text{'}}^{2}=8\phantom{\rule{0ex}{0ex}}5x{\text{'}}^{2}+5y{\text{'}}^{2}+3x{\text{'}}^{2}-3y{\text{'}}^{2}=8\phantom{\rule{0ex}{0ex}}8x{\text{'}}^{2}+2y{\text{'}}^{2}=8\phantom{\rule{0ex}{0ex}}x{\text{'}}^{2}+\frac{1}{4}y{\text{'}}^{2}=1\phantom{\rule{0ex}{0ex}}\frac{x{\text{'}}^{2}}{1}+\frac{y{\text{'}}^{2}}{4}=1\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

This represent equation of ellipse.