Rotate the Axes So That the New Equation Contains No $31 x^{2}+10 \sqrt{3} x y+21 y^{2}-144=0$

question 40

Multiple Choice

Rotate the axes so that the new equation contains no xy-term. Discuss the new equation.
- $31 x^{2}+10 \sqrt{3} x y+21 y^{2}-144=0$

• A)
• \[
• \begin{array}{l}
• \theta=45^{\circ} \\
• x^{\prime 2}=-4 \sqrt{2} y^{\prime}
• \end{array}
• \]
• parabola
• vertex at \( (0,0) \)
• focus at \( (0,-\sqrt{2}) \)
• B)
• \(\begin{array}{l}
• \theta=36.9^{\circ} \\
• \frac{x^{2}}{9}+\frac{y^{\prime 2}}{4}=1 \\
• \text { ellipse } \\
• \text { center at }(0,0) \\
• \text { major axis is } x^{\prime} \text {-axis } \\
• \text { vertices at }(\pm 3.0)
• \end{array}\)
• C)
• \[
• \begin{array}{l}
• \theta=30^{0} \\
• \frac{x^{2}}{4}+\frac{y^{+2}}{9}=1
• \end{array}
• \]
• ellipse
• center at \( (0,0) \)
• major axis is \( y^{\prime} \)-axi:
• vertices at \( (0, \pm 3) \)
• D)
• \(\begin{array}{l}
• \theta=45^{\circ} \\
• y^{\prime 2}=-4 \sqrt{2} x^{\prime} \\
• \text { parabola } \\
• \text { vertex at }(0,0) \\
• \text { focus at }(-\sqrt{2}, 0)
• \end{array}\)

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