Rotate the Axes So That the New Equation Contains No $x^{2}+2 x y+y^{2}-8 x+8 y=0$

question 91

Multiple Choice

Rotate the axes so that the new equation contains no xy-term. Discuss the new equation.
- $x^{2}+2 x y+y^{2}-8 x+8 y=0$

• A)
• \[
• \begin{array}{l}
• \theta=36.9^{0} \\
• \frac{x^{2}}{4}+\frac{y^{\prime 2}}{4}=1
• \end{array}
• \]
• ellipse
• center \( (0,0) \)
• major axis is \( x^{\prime} \)-axis
• vertices at \( (\pm 2,0) \)
• B)
• \[
• \begin{array}{l}
• \theta=36.9^{\circ} \\
• \frac{x^{2}}{4}+\frac{y^{2}}{2}=1
• \end{array}
• \]
• ellipse
• center \( (0,0) \)
• major axis is \( x^{4} \)-axi:
• vertices at \( (\pm 2,0) \)
• C)
• \[
• \begin{array}{l}
• \theta=45^{\circ} \\
• x^{\prime 2}=-4 \sqrt{2} y^{\prime}
• \end{array}
• \]
• parabola
• vertex at \( (0,0) \)
• focus at \( (0,-\sqrt{2}) \)
• D)
• \[
• \theta=45^{\circ}
• \]
• \[
• y^{\prime 2}=-4 \sqrt{2} x^{\prime}
• \]
• parabola
• vertex at \( (0,0) \)
• focus at \( (-\sqrt{2}, 0) \)

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