Question

Use the discriminant to determine whether the graph of the equation 13 x²+6 √3xy+7 y²=16 is a parabola, an ellipse, or a hyperbola. For 5 points extra credit, determine the angle of rotation. For another 5 points extra credit, convert the equation to an equation in the rotated coordinate system and sketch the graph.

Answer 1

The graph of the equation represents a hyperbola.

Step-by-step explanation:

To determine the type of conic section represented by the equation, we can look at the discriminant of the equation.

For the equation of the form Ax² + Bxy + Cy² + Dx + Ey + F = 0, the discriminant is given by B² - 4AC.

In this case, A = 13, B = 6√3, and C = 7. Substituting these values into the discriminant formula, we get:

B² - 4AC = (6√3)² - 4(13)(7) = 108 - 364 = -256.

Since the discriminant is negative, the graph of the equation represents a hyperbola.