\( 3 x ^ { 2 } + y ^ { 2 } + 18 x - 2 y - 8 = 0 \)

This equation describes a conic section. By rearranging and completing the square for both \( x \) and \( y \), you can identify its shape. Specifically, you'll find that it represents an ellipse. Ellipses are fascinating because they have unique properties, such as the fact that the sum of the distances from any point on the ellipse to the two foci is constant—perfect for imagining cosmic orbits! Often, people make the mistake of assuming all quadratic equations are parabolas. However, depending on the discriminant, you could be dealing with an ellipse, hyperbola, or parabola. Always check the coefficients to determine the type of conic section. Completing the square is your power move here!