Chapter 12, Problem 4CLT

To determine

Determine the number of degree of rotation.

Answer to Problem 4CLT

The angle is $\boxed{\theta =37.98°}$

Explanation of Solution

Given information:

Determine the number of degrees the axis must be rotated to eliminate the $xy$ -terms of the conic $x^2-4xy+2y^2=6$ Then sketch the conic.

Calculation:

Consider the given equation,

  $x^2-4xy+2y^2=6$

To identify conic type rearranges the equation in standard form,

  $x^2-4xy+2y^2=6$

Standard form is.

  $A{x}^2+C{y}^2+Dx+Ey+F=0$

  $x^2-4xy+2y^2=6$

On comparing with standard form the value of,

  $\begin{array}{l}A=1\\ B=-4\\ C=2\end{array}$

Now find the value of angle from standard form,

  $\cot 2\theta =\frac{A-C}{B}$

Hence angle is,

  $\begin{array}{l}\cot 2\theta =\frac{1-2}{-4}\\ \cot 2\theta =\frac{1}{4}\\ 2\theta ={\cot }^{-1}\left(\frac{1}{4}\right)\\ \theta =37.98°\end{array}$

Hence the angle is $\boxed{\theta =37.98°}$

To sketch the graph of equation with the transformation,

  $\begin{array}{l}x=X\cos \theta -Y\sin \theta \\ y=X\sin \theta +Y\cos \theta \\ x=0.788X-0.615Y\\ y=0.615X+0.788Y\end{array}$

Now replace the equation by above values.

  $\begin{array}{l}{\left(0.788X-0.615Y\right)}^2-4\left(0.788X-0.615Y\right)\left(0.615X+0.788Y\right)\\ +2{\left(0.615X+0.788Y\right)}^2=6\end{array}$

Simplify further,

  $\begin{array}{l}\left(0.621X^2-0.969XY+0.378Y^2\right)\\ -4\left(0.485X^2+0.621XY-0.378XY-0.485Y^2\right)\\ +2\left(0.378X^2+0.969XY+0.621Y^2\right)=6\\ X^2\left(0.621-1.94+0.756\right)-XY\left(0.969+0.972-1.938\right)+Y^2\left(0.378+0.485+0.621\right)=6\\ -0.563X^2+1.484Y^2=6\end{array}$

Express $Y$ with the standard quadratic equation,

  $x^2-4xy+2y^2=6$

  $a{y}^2+by+c=0$

  $\begin{array}{l}a=2\\ b=-4x\\ c=x^2-6\end{array}$

  $\begin{array}{l}y=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ =\frac{-\left(-4x\right)\pm \sqrt{16x^2-4\times 2\left(x^2-6\right)}}{2\times 2}\\ =\frac{\left(4x\right)\pm \sqrt{8x^2+48}}{4}\end{array}$

Use graphic utility $TI-83$ calculator to draw the graph.

Press $Y=$ and enter the equation. The display will be,

  

Now press window and select proper scale,

  

Now press graph.