Chapter A5.3, Problem 33E

To determine

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

Answer to Problem 33E

The new equation is a hyperbola.

The angle is $\alpha \approx {39.35}^{\circ }$ .

Sine and cosine angle is,

  $\begin{array}{l}\sin \alpha \approx 0.63\\ \cos \alpha \approx 0.77\end{array}$

Explanation of Solution

Given:

  $3x^2+5xy+2y^2-8y-1=0$

Calculation:

Comparing with the standard equation for a quadratic curve,

  $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$

Thus,

  $A=3,B=5,C=2$

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

  $\begin{array}{l}\tan 2\alpha =\frac{B}{A-C}\\ =\frac{5}{3-2}\\ =5\end{array}$

Now, using calculator calculate the angle .

  $\begin{array}{l}2\alpha \approx {78.69}^{\circ }\\ \Rightarrow \alpha \approx {39.35}^{\circ }\end{array}$

And,

  $\begin{array}{l}\sin \alpha \approx 0.63\\ \cos \alpha \approx 0.77\end{array}$

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

  $\begin{array}{l}x=x^{\text{'}}\cos \alpha -y^{\text{'}}\sin \alpha \\ =0.77x^{\text{'}}-0.63y^{\text{'}}\\ y=x^{\text{'}}\sin \alpha +y^{\text{'}}\cos \alpha \\ =0.63x^{\text{'}}+0.77y^{\text{'}}\end{array}$

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

  $\begin{array}{l}3{\left(0.77x^{\text{'}}-0.63y^{\text{'}}\right)}^2+5\left(0.77x^{\text{'}}-0.63y^{\text{'}}\right)\left(0.63x^{\text{'}}+0.77y^{\text{'}}\right)+2{\left(0.63x^{\text{'}}+0.77y^{\text{'}}\right)}^2\\ -8\left(0.63x^{\text{'}}+0.77y^{\text{'}}\right)-1=0\end{array}$

Simplify.

  $5.05x^{\text{'}}{}^2-0.05y^{\text{'}}{}^2-5.07x^{\text{'}}-6.19y^{\text{'}}-1=0$

Hence, the new equation is a hyperbola, having coefficients.

  $A^{\text{'}}=5.05,B^{\text{'}}=0.00,C^{\text{'}}=-0.05,D^{\text{'}}=-5.07,E^{\text{'}}=-6.19,F^{\text{'}}=-1$

Conclusion:

The new equation is a hyperbola.

The angle is $\alpha \approx {39.35}^{\circ }$ .

Sine and cosine angle is,

  $\begin{array}{l}\sin \alpha \approx 0.63\\ \cos \alpha \approx 0.77\end{array}$