Rotate the Axes So That the New Equation Contains No $x y + 16 = 0$

question 131

Multiple Choice

Rotate the axes so that the new equation contains no xy-term. Discuss the new equation.
- $x y + 16 = 0$

• A) \[
• \theta=45^{\circ}
• \]
• \[
• \frac{y^{\prime 2}}{32}+\frac{x^{\prime 2}}{32}=1
• \]
• ellipse
• center at \( (0,0) \)
• major axis is \( y^{\prime} \)-axis
• vertices at \( (0, \pm 4 \sqrt{2}) \)
• B) \[
• \theta=45^{\circ}
• \]
• \[
• \frac{y^{\prime 2}}{32}-\frac{x^{\prime 2}}{32}=1
• \]
• hyperbola
• center at \( (0,0) \)
• transverse axis is \( \mathrm{y}^{\prime} \)-axis
• vertices at \( (0, \pm 4 \sqrt{2}) \)
• C) \(\begin{array}{l}
• \theta=45^{\circ} \\
• y^{\prime 2}=-32 x^{\prime} \\
• \text { parabola } \\
• \text { vertex at }(0,0) \\
• \text { focus at }(-8,0)
• \end{array}\)
• D) \[
• \begin{array}{l}
• \theta=36.9^{0} \\
• \frac{x^{12}}{4}+\frac{y^{\prime 2}}{2}=1
• \end{array}
• \]
• ellipse
• center at \( (0,0) \)
• major axis is the \( x^{\prime} \)-axis
• vertices at \( (\pm 2,0) \)

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