Rotate the Axes So That the New Equation Contains No $24 x y - 7 y ^ { 2 } + 36 = 0$

question 170

Multiple Choice

Rotate the axes so that the new equation contains no xy-term. Discuss the new equation.
- $24 x y - 7 y ^ { 2 } + 36 = 0$

• A)\[
• \begin{array}{l}
• \theta=36.9^{\circ} \\
• \frac{y^{\prime 2}}{9}-\frac{x^{\prime 2}}{16}=1
• \end{array}
• \]
• hyperbola
• center at \( (0,0) \)
• transverse axis is the \( y^{\prime} \)-axis
• vertices at \( (0, \pm 3) \)
• B)\[
• \begin{array}{l}
• \theta=36.9^{\circ} \\
• \frac{4 y^{\prime 2}}{9}-\frac{x^{\prime 2}}{4}=1
• \end{array}
• \]
• hyperbola
• center at \( (0,0) \)
• transverse axis is the \( y^{\prime} \)-axis
• vertices at \( \left(0, \pm \frac{3}{2}\right) \)
• C) \[
• \begin{array}{l}
• \theta=36.9^{\circ} \\
• \frac{y^{\prime 2}}{4}-\frac{4 x^{2}}{9}=1
• \end{array}
• \]
• hyperbola
• center at \( (0,0) \)
• transverse axis is the \( y^{\prime} \)-axis
• vertices at \( (0, \pm 2) \)
• D)\[
• \begin{array}{l}
• \theta=53.1^{\circ} \\
• \frac{y^{\prime 2}}{4}-\frac{4 x^{\prime 2}}{9}=1
• \end{array}
• \]
• hyperbola
• center at \( (0,0) \)
• transverse axis is the \( y^{\prime} \)-axis
• vertices at \( (0, \pm 2) \)

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