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

question 36

Multiple Choice

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

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