Use the Discriminant to Determine If the Graph of the Equation

Essay

Use the discriminant to determine if the graph of the equation $\frac { 3 } { 4 } x ^ { 2 } + \frac { \sqrt { 3 } } { 2 } x y + \frac { 1 } { 4 } y ^ { 2 } + 4 x - 4 \sqrt { 3 } y = 0$ is a parabola, an ellipse or a hyperbola, and then use a rotation of axes to eliminate the $x y$ - term, and sketch the graph.