Rotate the Axes So That the New Equation Contains No $$x ^ { 2 } + x y + y ^ { 2 } - 3 y - 6 = 0$$

question 100

Multiple Choice

Rotate the axes so that the new equation contains no xy-term. Discuss the new equation.
- $$x ^ { 2 } + x y + y ^ { 2 } - 3 y - 6 = 0$$

• A)
• \(\begin{array}{l}
• \theta=45^{\circ} \\
• y^{\prime 2}=-18 x^{\prime} \\
• \text { parabola } \\
• \text { vertex at }(0,0) \\
• \text { focus at }\left(-\frac{9}{2}, 0\right)
• \end{array}\)
• B)
• \[
• \frac{\theta=45^{\circ}}{\left(x^{\prime}-\frac{\sqrt{2}}{2}\right)^{2}}+\frac{\left(y^{\prime}-\frac{3 \sqrt{2}}{2}\right)^{2}}{15}=1
• \]
• ellipse center at \( \left(\frac{\sqrt{2}}{2}, \frac{3 \sqrt{2}}{2}\right) \)
• major axis is \( y^{\prime} \)-axis vertices at \( \left(\frac{\sqrt{2}}{2},-\frac{3 \sqrt{2}}{2}\right) \) and \( \left(\frac{\sqrt{2}}{2}, \frac{9 \sqrt{2}}{2}\right) \)
• C) \[
• \theta=45^{\circ}
• \]
• \[
• \frac{x^{\prime 2}}{6}-\frac{y^{\prime 2}}{8}=1
• \]
• hyperbola
• center at \( (0,0) \)
• transverse axis is the \( \mathrm{x}^{\prime} \)-axis
• vertices at \( (\pm \sqrt{6}, 0) \)
• D)\[
• \begin{array}{l}
• \theta=45^{\circ} \\
• \frac{x^{\prime 2}}{3}+\frac{y^{2}}{4}=1
• \end{array}
• \]
• ellipse
• center at \( (0,0) \)
• major axis is \( y^{\prime} \)-axis
• vertices at \( (0, \pm 2) \)

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