Question

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.$13x^2-6\sqrt{3}xy+7y^2-16=0$

Short answer

The ellipse is rotated through an angle of${60}^{\circ }.$

$$\begin{gathered} x=\frac{1}{2}x\text{'}-\frac{\sqrt{3}}{2}y\text{'} \\ y=\frac{\sqrt{3}}{2}x\text{'}+\frac{1}{2}y\text{'} \end{gathered}$$

Step-by-step solution

Step 1. Given Information

The given equation is $13x^2-6\sqrt{3}xy+7y^2-16=0$.

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

Step 2. Finding acute angle

Here$A=13,B=-6\sqrt{3},C=7$

role="math" localid="1646931475071" $$\begin{gathered} cot2\theta =\frac{A-C}{B}=\frac{13-7}{-6\sqrt{3}} \\ cot2\theta =\frac{6}{-6\sqrt{3}}=-\frac{\sqrt{3}}{3} \end{gathered}$$

$\theta$need to be acute.

So, $$\begin{gathered} cot2\theta =-\frac{\sqrt{3}}{3} \\ 2\theta =120 \\ \theta ={60}^{\circ } \end{gathered}$$

Step 3. Finding the values of x,y

$$\begin{gathered} x=x\text{'}\cos {60}^{\circ }-y\text{'}\sin {60}^{\circ } \\ x=\frac{1}{2}x\text{'}-\frac{\sqrt{3}}{2}y\text{'} \end{gathered}$$

Similarly,$y=\frac{\sqrt{3}}{2}x\text{'}+\frac{1}{2}y\text{'}$

Step 4. Substitute these values in the original equation

$13x^2-6\sqrt{3}xy+7y^2-16=0$

$$\begin{gathered} \frac{13}{4}{\left(x\text{'}-\sqrt{3}y\text{'}\right)}^2-\frac{6\sqrt{3}}{2\times 2}\left(x\text{'}-\sqrt{3}y\text{'}\right)\left(\sqrt{3}x\text{'}+y\text{'}\right)+\frac{7}{4}{\left(\sqrt{3}x\text{'}+y\text{'}\right)}^2-16=0 \\ \frac{13}{4}\left(x{\text{'}}^2-2\sqrt{3}x\text{'}y\text{'}+3y{\text{'}}^2\right)-\frac{3\sqrt{3}}{2}\left(\sqrt{3}x{\text{'}}^2+x\text{'}y\text{'}-3x\text{'}y\text{'}+\sqrt{3}y{\text{'}}^2\right)+\frac{7}{4}\left(3x{\text{'}}^2+2\sqrt{3}x\text{'}y\text{'}+y{\text{'}}^2\right)=16 \end{gathered}$$

Step 5. Simplify the above Expression

$$\begin{gathered} \frac{13}{4}x{\text{'}}^2+\frac{39}{4}y{\text{'}}^2-\frac{13\sqrt{3}}{2}x\text{'}y\text{'}-\frac{9}{2}x{\text{'}}^2-\frac{3\sqrt{3}}{2}x\text{'}y\text{'}+\frac{9\sqrt{3}}{2}x\text{'}y\text{'}-\frac{9}{2}y{\text{'}}^2+\frac{21}{4}x{\text{'}}^2+\frac{7}{4}y{\text{'}}^2+\frac{7\sqrt{3}}{2}x\text{'}y\text{'}-16=0 \\ \frac{13}{4}x{\text{'}}^2-\frac{9}{2}x{\text{'}}^2+\frac{39}{4}y{\text{'}}^2-\frac{9}{2}y{\text{'}}^2-\frac{13\sqrt{3}}{2}x\text{'}y\text{'}-\frac{3\sqrt{3}}{2}x\text{'}y\text{'}+\frac{9\sqrt{3}}{2}x\text{'}y+\frac{7\sqrt{3}}{2}x\text{'}y\text{'}=16 \\ \frac{5}{4}x{\text{'}}^2+\frac{21}{4}y{\text{'}}^2=16 \end{gathered}$$

Step 6. Graphing Equation