For the​ equation, x squared plus 4 xy minus 2 y squared minus 2equals​0, answer the following questions.
a. Rewrite the equation in a rotated xprimeyprime​-system without an xprimeyprime​-term. Use the appropriate rotation formulas.
b. Express the equation involving xprime and yprime in the standard form of a conic section.
c. Use the rotated system to graph the system.

Given the equation \( x^2 + 4xy - 2y^2 - 2 = 0 \), we need to:

a. Rewrite the equation in a rotated \(x'y'\)-system without an \(x'y'\)-term.
b. Express the equation involving \(x'\) and \(y'\) in the standard form of a conic section.
c. Use the rotated system to graph the system.

### Part a: Rotating the Coordinate System

To eliminate the \(xy\)-term, we can use the rotation formulas. The angle \(\theta\) for the rotation is given by:
\[
\tan 2\theta = \frac{B}{A - C}
\]
where the original equation is \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\).

In our case:
- \(A = 1\)
- \(B = 4\)
- \(C = -2\)
- \(D = 0\)
- \(E = 0\)
- \(F = -2\)

So, we have:
\[
\tan 2\theta = \frac{4}{1 - (-2)} = \frac{4}{3}
\]
Solving for \(\theta\):
\[
2\theta = \tan^{-1}\left(\frac{4}{3}\right)
\]
\[
\theta = \frac{1}{2}\tan^{-1}\left(\frac{4}{3}\right)
\]

Let's calculate \(\theta\):
\[
\theta \approx \frac{1}{2} \times 53.13^\circ \approx 26.57^\circ
\]

Using this angle, we apply the rotation formulas:
\[
x = x'\cos\theta - y'\sin\theta
\]
\[
y = x'\sin\theta + y'\cos\theta
\]

Substituting these into the original equation \(x^2 + 4xy - 2y^2 - 2 = 0\) will result in an equation without the \(x'y'\)-term.

### Part b: Standard Form of the Conic Section

Substituting \(x = x'\cos\theta - y'\sin\theta\) and \(y = x'\sin\theta + y'\cos\theta\) into the equation \(x^2 + 4xy - 2y^2 - 2 = 0\):

\[
(x'\cos\theta - y'\sin\theta)^2 + 4(x'\cos\theta - y'\sin\theta)(x'\sin\theta + y'\cos\theta) - 2(x'\sin\theta + y'\cos\theta)^2 - 2 = 0
\]

Expanding this and simplifying should give us an equation in terms of \(x'\) and \(y'\) without the \(x'y'\)-term. After simplification, we will get the standard form.

### Part c: Graphing the System

Once we have the standard form of the conic section, we can graph it in the rotated coordinate system.

Let's perform the necessary calculations.

First, we'll substitute the rotation:
\[
\cos \theta = \cos 26.57^\circ \approx 0.894
\]
\[
\sin \theta = \sin 26.57^\circ \approx 0.447
\]

Now, substitute into the original equation:
\[
(x' \cdot 0.894 - y' \cdot 0.447)^2 + 4(x' \cdot 0.894 - y' \cdot 0.447)(x' \cdot 0.447 + y' \cdot 0.894) - 2(x' \cdot 0.447 + y' \cdot 0.894)^2 - 2 = 0
\]

Expanding and combining like terms gives us:
\[
(x'^2 \cdot 0.894^2 + 2x'y' \cdot 0.894 \cdot -0.447 + y'^2 \cdot -0.447^2) + 4(x'^2 \cdot 0.894 \cdot 0.447 + x'y' \cdot 0.894 \cdot 0.894 - y'^2 \cdot 0.447 \cdot 0.447) - 2(x'^2 \cdot 0.447^2 + 2x'y' \cdot 0.447 \cdot 0.894 + y'^2 \cdot 0.894^2) - 2 = 0
\]

Simplify the coefficients to eliminate \(x'y'\) term:
\[
(0.799x'^2 - 0.399y'^2) + 4(0.399x'^2 - 0.199y'^2) - 2(0.199x'^2 + 0.799y'^2) - 2 = 0
\]

Combining like terms:
\[
(0.799 + 4 \cdot 0.399 - 2 \cdot 0.199)x'^2 + (-0.399 + 4 \cdot -0.199 - 2 \cdot 0.799)y'^2 - 2 = 0
\]

Calculating coefficients:
\[
(0.799 + 1.596 - 0.398)x'^2 + (-0.399 - 0.796 - 1.598)y'^2 - 2 = 0
\]
\[
(1.997)x'^2 + (-2.793)y'^2 - 2 = 0
\]

Divide through by 2:
\[
\approx x'^2 - \frac{3}{2} y'^2 - 1 = 0
\]

This is the equation of the conic section in the rotated coordinate system. For the graph, we can now plot this transformed equation in \(x'\)-\(y'\) coordinates.

If you have any questions or need further details, please let me know.

Here are 5 questions you can ask next:
1. Can you explain how to derive the rotation formulas for conic sections?
2. How do you determine the type of conic section from its general equation?
3. What is the significance of the discriminant in identifying conic sections?
4. How do you convert a conic section from general form to standard form?
5. Can you provide examples of real-world applications of conic sections?

**Tip:** When rotating conic sections, ensure you correctly calculate the angle \(\theta\) to accurately eliminate the cross-product term \(xy\).

Explore how to rewrite conic section equations using rotated coordinates without cross-product terms. Learn to graph conic sections in rotated systems with detailed examples and step-by-step solutions. Suitable for advanced high school students and beyond.

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