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

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

A)
$\begin{array}{l}\theta=36.9^{\circ} \\\frac{x^{\prime 2}}{4}+\frac{y^{\prime 2}}{4}=1 \\\text { ellipse } \\\text { center }(0,0) \\\text { major axis is } x^{\prime} \text {-axi: } \\\text { vertices at }(\pm 2,0) \end{array}$

B)
$\begin{array}{l}\theta=36.9^{\circ} \\\frac{x^{2}}{4}+\frac{y^{2}}{2}=1 \\\text { ellipse } \\\text { center }(0,0) \\\text { major axis is } x^{\prime} \text {-axi: } \\\text { vertices at }(\pm 2,0) \end{array}$

C)
$\begin{array}{l}\theta=45^{\circ} \\x^{\prime 2}=-4 \sqrt{2} y^{\prime} \\\text { parabola } \\\text { vertex at }(0,0) \\\text { focus at }(0,-\sqrt{2}) \end{array}$

D)
$\begin{array}{l}\theta=45^{\circ} \\y^{\prime 2}=-4 \sqrt{2} x^{\prime} \\\text { parabola } \\\text { vertex at }(0,0) \\\text { focus at }(-\sqrt{2}, 0) \end{array}$

Correct Answer:

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