Chapter A5.3, Problem 34E

To determine

To find: a sine and cosine angle and say whether the equation is an ellipse, hyperbola or parabola.

Answer to Problem 34E

The new equation is an ellipse.

The angle is $\alpha \approx -{22.5}^{\circ }$ .

Sine and cosine angle is,

  $\begin{array}{l}\sin \alpha \approx -0.38\\ \cos \alpha \approx 0.92\end{array}$

Explanation of Solution

Given:

  $2x^2+7xy+9y^2+20x-86=0$

Calculation:

Comparing with the standard equation for a quadratic curve,

  $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$

Thus,

  $A=2,B=7,C=9$

For the cross-product term to vanish, the coordinate axes must be rotated by an angle $\alpha$ , such that ,

  $\begin{array}{l}\tan 2\alpha =\frac{B}{A-C}\\ =\frac{7}{2-9}\\ =-1\end{array}$

Now, using calculator calculate the angle,

  $\begin{array}{l}2\alpha \approx -{45}^{\circ }\\ \Rightarrow \alpha \approx -{22.5}^{\circ }\end{array}$

And,

  $\begin{array}{l}\sin \alpha \approx -0.38\\ \cos \alpha \approx 0.92\end{array}$

After rotation of coordinate axes, the original coordinates, $x$ and $y$ , are replaced by new coordinates $x^{\text{'}}$ and $y^{\text{'}}$ , such that,

  $\begin{array}{l}x=x^{\text{'}}\cos \alpha -y^{\text{'}}\sin \alpha \\ =0.92x^{\text{'}}+0.38y^{\text{'}}\\ y=x^{\text{'}}\sin \alpha +y^{\text{'}}\cos \alpha \\ =-0.38x^{\text{'}}-0.92y^{\text{'}}\end{array}$

Substituting the above values of $x$ and $y$ in the given quadratic equation,

  $\begin{array}{l}2{\left(0.92x^{\text{'}}-0.38y^{\text{'}}\right)}^2+7\left(0.92x^{\text{'}}+0.38y^{\text{'}}\right)\left(-0.38x^{\text{'}}-0.92y^{\text{'}}\right)+9{\left(-0.38x^{\text{'}}-0.92y^{\text{'}}\right)}^2\\ +20\left(0.92x^{\text{'}}+0.38y^{\text{'}}\right)-86=0\end{array}$

Simplify,

  $0.55{\left(x^{\text{'}}\right)}^2+10.45{\left(y^{\text{'}}\right)}^2+18.48x^{\text{'}}+7.65y^{\text{'}}-86=0$

Hence, the new equation is an ellipse, having coefficients,

  $A^{\text{'}}=0.55,B^{\text{'}}=0.00,C^{\text{'}}=10.45,D^{\text{'}}=18.48,E^{\text{'}}=7.65,F^{\text{'}}=-86$

Conclusion:

The new equation is an ellipse.

The angle is $\alpha \approx -{22.5}^{\circ }$ .

Sine and cosine angle is,

  $\begin{array}{l}\sin \alpha \approx -0.38\\ \cos \alpha \approx 0.92\end{array}$