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

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 $(0, \pm 2)$