Refer to the equation A x^2+C y^2+D x+E y+F=0, where A and C...

refer to the equation $A x^2+C y^2+D x+E y+F=0$, where $A$ and $C$ are not both 0 .
In Exercise 42 , why must $A\left(4 A C F-C D^2-A E^2\right)$ be negative? What can be said about the graph if $A\left(4 A C F-C D^2-A E^2\right)$ is 0 ? What if it is positive?

Key Concepts

General Equation of a Conic Section

Conic sections in the plane can be represented by a general second?degree equation of the form Ax² + Bxy + Cy² + Dx + Ey + F = 0. In cases where the xy-term (B) is missing, the equation takes the simpler form Ax² + Cy² + Dx + Ey + F = 0. This representation is central to understanding how changing coefficients affect the shape, location, and type (ellipse, parabola, hyperbola) of the curve.

Completing the Square

Completing the square is a technique used to rewrite quadratic expressions in a form that makes the geometry of the conic clear. For the equation Ax² + Cy² + Dx + Ey + F = 0, one can complete the square in x and y separately to obtain expressions of the form A(x - h)² and C(y - k)². This process reveals the center and the squared radii of the conic and helps identify conditions under which the equation represents a real, nondegenerate conic.

Nondegeneracy Conditions

For a conic section to be nondegenerate (i.e. to represent a curve rather than a point, a pair of lines, or the empty set), the terms obtained after completing the square must allow a positive constant on the other side of the equation. In the specific process for Ax² + Cy² + Dx + Ey + F = 0, the resulting constant involves the expression 4ACF - C·D² - A·E². When multiplied by A, requiring A(4ACF - C·D² - A·E²) to be negative ensures that the sum of the squared terms equals a positive number, which is necessary for the existence of a real conic (typically an ellipse when A and C have the same sign).

Degenerate Cases and the Sign of the Invariant

The sign of the invariant A(4ACF - C·D² - A·E²) indicates whether or not the conic is degenerate. If this expression equals 0, the completed square yields a zero on the right-hand side, meaning that the graph reduces to a single point (a degenerate ellipse). If the expression is positive, then the squared terms would have to equal a negative number, which is impossible in the realm of real numbers; hence, the equation has no real graph. Thus, the negativity of the expression is exactly what is required for the conic to be a nondegenerate, real ellipse.

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