Use the interactive to describe the conic given by the equation 3x2 – 9xy + 8y2 + x + y – 27 = 0.

The quadratic-form matrix is [[3, −4.5], [−4.5, 8]]. Its determinant is 3.75 > 0 and its eigenvalues are positive, so the conic is an ellipse. Solving the equations for its center gives (−5/3, −1). The major-axis direction has positive slope, and the foci lie on that axis, on opposite sides of the center; they fall in the first and third quadrants. Thus the description is “Ellipse with foci in the first and third quadrants.”

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