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.$11x^2+10\sqrt{3}xy+y^2-4=0$

Short answer

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

Step-by-step solution

Step 1. Given Information

The given equation is $11x^2+10\sqrt{3}xy+y^2-4=0$.

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

Step 2. Finding acute angle 

Here $A=11,B=10\sqrt{3},C=1$

$$\begin{gathered} cot2\theta =\frac{11-1}{10\sqrt{3}}=\frac{12}{10\sqrt{3}} \\ cot2\theta =\frac{\sqrt{3}}{3} \end{gathered}$$

$\theta$need to be acute.

$$\begin{gathered} cot2\theta =\frac{\sqrt{3}}{3} \\ 2\theta ={60}^{\circ } \\ \theta ={30}^{\circ } \end{gathered}$$

Step 3. Finding expression for x,y

$$\begin{gathered} x=x\text{'}\cos \left({30}^{\circ }\right)-y\text{'}\sin \left({30}^{\circ }\right) \\ x=\frac{\sqrt{3}}{2}x\text{'}-\frac{1}{2}y\text{'} \\ and \\ y=x\text{'}\sin \left({30}^{\circ }\right)+y\text{'}\cos \left({30}^{\circ }\right) \\ y=\frac{1}{2}x\text{'}+\frac{\sqrt{3}}{2}y\text{'} \end{gathered}$$

Step 4. Substitute these values in the original Equation

$$\begin{gathered} 11x^2+10\sqrt{3}xy+y^2-4=0 \\ \frac{11}{4}{\left(\sqrt{3}x\text{'}-y\text{'}\right)}^2+\frac{10\sqrt{3}}{2\times 2}\left(\sqrt{3}x\text{'}-y\text{'}\right)\left(x\text{'}+\sqrt{3}y\text{'}\right)+\frac{1}{4}{\left(x\text{'}+\sqrt{3}y\text{'}\right)}^2-4=0 \end{gathered}$$

Step 5. Simplify the above Expression

$$\begin{gathered} \frac{11}{4}\left(3x{\text{'}}^2-2\sqrt{3}x\text{'}y\text{'}+y{\text{'}}^2\right)+\frac{5\sqrt{3}}{2}\left(\sqrt{3}x{\text{'}}^2+3x\text{'}y\text{'}-x\text{'}y\text{'}-\sqrt{3}y{\text{'}}^2\right)+\frac{1}{4}\left(x{\text{'}}^2+2\sqrt{3}x\text{'}y\text{'}+3y{\text{'}}^2\right)=4 \\ \frac{33}{4}x{\text{'}}^2-\frac{11\sqrt{3}}{2}x\text{'}y\text{'}+\frac{11}{4}y{\text{'}}^2+\frac{15}{2}x{\text{'}}^2+\frac{15\sqrt{3}}{2}x\text{'}y\text{'}-\frac{5\sqrt{3}}{2}x\text{'}y\text{'}-\frac{15}{2}y{\text{'}}^2+\frac{1}{4}x{\text{'}}^2+\frac{\sqrt{3}}{2}x\text{'}y\text{'}+\frac{3}{4}y{\text{'}}^2=4 \\ \frac{33}{4}x{\text{'}}^2+\frac{15}{2}x{\text{'}}^2+\frac{1}{4}x{\text{'}}^2+\frac{11}{4}y{\text{'}}^2-\frac{15}{2}y{\text{'}}^2+\frac{3}{4}y{\text{'}}^2-\frac{11\sqrt{3}}{2}x\text{'}y\text{'}+\frac{15\sqrt{3}}{2}x\text{'}y\text{'}-\frac{5\sqrt{3}}{2}x\text{'}y\text{'}+\frac{\sqrt{3}}{2}x\text{'}y\text{'}=4 \\ 16x{\text{'}}^2-16y{\text{'}}^2=4 \\ 4x{\text{'}}^2-4y{\text{'}}^2=1 \end{gathered}$$

Step 6. Graphing Equation