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

Short answer

The ellipse is rotated through an angle of$-33.{69}^{\circ }$and equation is$46.2x{\text{'}}^2+18.6y{\text{'}}^2-9.16x\text{'}y\text{'}-14.3\sqrt{13}x\text{'}-13.2\sqrt{13}y\text{'}=0$

$$\begin{gathered} x=0.8321x\text{'}+0.5527y\text{'} \\ y=-0.5527x\text{'}+0.8321y\text{'} \end{gathered}$$

Step-by-step solution

Step 1. Given Information

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

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

Step 2. Finding acute angle

Here $A=25,B=-36,C=40$

$$\begin{gathered} cot2\theta =\frac{25-40}{-36}=\frac{5}{12} \\ 2\theta =-67.38 \\ \theta =-33.{69}^{\circ } \end{gathered}$$

Step 3. Finding expression for x,y

$$\begin{gathered} x=x\text{'}\cos \left(-33.69\right)-y\text{'}\sin \left(-33.69\right) \\ x=0.8321x\text{'}+0.5527y\text{'} \\ and \\ y=x\text{'}\sin \left(-33.69\right)+y\text{'}\cos \left(-33.69\right) \\ y=-0.5547x\text{'}+0.8321y\text{'} \end{gathered}$$

Step 5. Substitute these values in the original Equation

$$\begin{gathered} 25x^2-36xy+40y^2-12\sqrt{13}x-8\sqrt{13}y=0 \\ 25{\left(0.8321x\text{'}+0.5547y\text{'}\right)}^2-36\left(0.8321x\text{'}+0.5547y\text{'}\right)\left(-0.5547x\text{'}+0.8321y\text{'}\right)+40{\left(-0.5547x\text{'}+0.8321y\text{'}\right)}^2-12\sqrt{13}\left(0.8321x\text{'}+0.5547y\text{'}\right)-8\sqrt{13}\left(-0.5547x\text{'}+0.8321y\text{'}\right)=0 \end{gathered}$$

Step 5. Simplify the above Equation

$$\begin{gathered} 17.3x{\text{'}}^2+23.07x\text{'}y\text{'}+7.6y{\text{'}}^2+16.6x{\text{'}}^2-24.9x\text{'}y\text{'}+11.07x\text{'}y\text{'}-16.6y{\text{'}}^2+27.6y{\text{'}}^2-18.4x\text{'}y\text{'}+12.3x{\text{'}}^2-9.9\sqrt{13}x\text{'}-6.6\sqrt{13}y\text{'}-4.4\sqrt{13}x\text{'}-6.6\sqrt{13}y\text{'}=0 \\ 46.2x{\text{'}}^2+18.6y{\text{'}}^2-9.16x\text{'}y\text{'}-14.3\sqrt{13}x\text{'}-13.2\sqrt{13}y\text{'}=0 \end{gathered}$$

Step 6. Graphing Equation