Use the Discriminant to Classify the Graph; Then Use the Quadratic

Use the discriminant to classify the graph; then use the quadratic formula to solve for $y$ .
$$3 x ^ { 2 } - 2 \sqrt { 3 } x y + y ^ { 2 } + 10 \sqrt { 3 } x - 6 y - 24 = 0$$

A) ellipse; $\quad y = 3 + \sqrt { 3 } x \pm \sqrt { 4 \sqrt { 3 } x + 33 }$
B) ellipse; $\quad y = 3 + \sqrt { 3 } x \pm \sqrt { - 4 \sqrt { 3 } x + 33 }$
C) parabola; $y = 3 + \sqrt { 3 } x \pm \sqrt { - 4 \sqrt { 3 } x + 33 }$
D) parabola; $y = \sqrt { 3 } x \pm \sqrt { - 4 \sqrt { 3 } x + 33 }$
E) hyperbola; $y = \sqrt { 3 } x \pm \sqrt { - 4 \sqrt { 3 } x + 33 }$