1. **State the problem:** We need to determine the nature of the conic given by the equation $$x^2 - 2xy - 12y - 2y^2 + 6x = 20$$. 2. **Rewrite the equation:** Bring all terms to one side: $$x^2 - 2xy - 2y^2 + 6x - 12y - 20 = 0$$ 3. **Identify coefficients:** The general second-degree equation is $$Ax^2 + 2Bxy + Cy^2 + 2Dx + 2Ey + F = 0$$. Here, comparing: - $$A = 1$$ - $$2B = -2 \Rightarrow B = -1$$ - $$C = -2$$ - $$2D = 6 \Rightarrow D = 3$$ - $$2E = -12 \Rightarrow E = -6$$ - $$F = -20$$ 4. **Calculate the discriminant:** $$\Delta = B^2 - AC = (-1)^2 - (1)(-2) = 1 + 2 = 3$$ 5. **Interpret the discriminant:** - If $$\Delta > 0$$, the conic is a hyperbola. - If $$\Delta = 0$$, the conic is a parabola. - If $$\Delta < 0$$, the conic is an ellipse (or circle if $$A=C$$ and $$B=0$$). Since $$\Delta = 3 > 0$$, the conic is a **hyperbola**. **Final answer:** The conic $$x^2 - 2xy - 12y - 2y^2 + 6x = 20$$ represents a hyperbola.