Rotate the Axes So That the New Equation Contains No $$5 x ^ { 2 } - 6 x y + 5 y ^ { 2 } - 8 = 0$$

question 109

Multiple Choice

Rotate the axes so that the new equation contains no xy-term. Discuss the new equation.
- $$5 x ^ { 2 } - 6 x y + 5 y ^ { 2 } - 8 = 0$$

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

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Definitions:

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