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.$16x^2+24xy+9y^2-130x+90y=0$

Short answer

The parabola is rotated through an angle of$-36.{87}^{\circ }$

Step-by-step solution

Step 1. Given Information

The given equation is$16x^2+24xy+9y^2-130x+90y=0$.

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

Step 2. Finding acute angle

Here $A=16,B=24,C=9$

localid="1646936800893" $$\begin{gathered} cot2\theta =\frac{16-9}{24}=\frac{7}{24} \\ 2\theta =-73.74 \\ \theta =-36.87 \end{gathered}$$

Step 3. Finding expression for x,y

$$\begin{gathered} x=x\text{'}\cos \left(-36.87\right)-y\text{'}\sin \left(-36.87\right) \\ x=0.79x\text{'}+0.6y\text{'} \\ and \\ y=x\text{'}\sin \left(-36.87\right)+y\text{'}\cos \left(-36.87\right) \\ y=-0.6x\text{'}+0.79y\text{'} \end{gathered}$$

Step 4.  Substitute these values in the original equation

$$\begin{gathered} 16x^2+24xy+9y^2-130x+90y=0 \\ 16{\left(0.79x\text{'}+0.6y\text{'}\right)}^2+24\left(0.79x\text{'}+0.6y\text{'}\right)\left(-0.6x\text{'}+0.79y\text{'}\right)+9{\left(-0.6x\text{'}+0.79y\text{'}\right)}^2-130\left(0.79x\text{'}+0.6y\text{'}\right)+90\left(-0.6x\text{'}+0.79y\text{'}\right)=0 \\ 16\left(0.6x{\text{'}}^2+0.9x\text{'}y\text{'}+0.3y{\text{'}}^2\right)+24\left(-0.4x{\text{'}}^2+0.6x\text{'}y\text{'}-0.3x\text{'}y\text{'}+0.4y{\text{'}}^2\right)+9\left(0.6y{\text{'}}^2-0.9x\text{'}y\text{'}+0.3x{\text{'}}^2\right)-102.7x\text{'}-78y\text{'}-54x\text{'}+71.1y\text{'}=0 \end{gathered}$$

Step 5. Simplify above Equation

$$\begin{gathered} 9.6x{\text{'}}^2+14.4x\text{'}y\text{'}+4.8y{\text{'}}^2-9.6x{\text{'}}^2+7.2x\text{'}y\text{'}+9.6y{\text{'}}^2+5.4y{\text{'}}^2-8.1x\text{'}y\text{'}+2.7x{\text{'}}^2-102.7x\text{'}-78y\text{'}-54x\text{'}+71.1y\text{'}=0 \\ 2.7x{\text{'}}^2+13.5x\text{'}y\text{'}+19.8y{\text{'}}^2-156x\text{'}-6.9y\text{'}=0 \end{gathered}$$

Step 6. Graphing Equation