Use the Discriminant Test to determine the type of the conic...

Use the Discriminant Test to determine the type of the conic section (in each case, the equation is nondegenerate). Plot the curve if you have a computer algebra system.
$$
2 x^{2}-3 x y+5 y^{2}-4=0
$$

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Key Concepts

Discriminant Test

The Discriminant Test utilizes the expression B² - 4AC, where A, B, and C are the coefficients from the general quadratic equation of a conic section. This test determines the type of conic: a negative discriminant indicates an ellipse (or circle as a special case), zero indicates a parabola, and a positive discriminant signals a hyperbola.

Conic Sections

Conic sections are the curves obtained by slicing a double-napped cone with a plane. They include ellipses, parabolas, and hyperbolas, and their properties are often analyzed using the general quadratic equation in two variables. These curves play a crucial role in various fields such as physics, engineering, and astronomy.

General Quadratic Equation

The general quadratic equation for conic sections is given by Ax² + Bxy + Cy² + Dx + Ey + F = 0. By examining the coefficients, particularly through the discriminant B² - 4AC, one can classify and study the conic's geometric properties without necessarily resorting to completing the square or other methods.

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