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 43, why must $4 A C F-C D^2-A E^2$ not equal zero? What can be said about the graph if $4 A C F-C D^2-A E^2=0$ ?

Key Concepts

Non-Degenerate Conic Section

A non-degenerate conic section is one that truly represents an ellipse, hyperbola, or parabola rather than collapsing into a simpler figure. In the general quadratic equation, certain conditions must be met to ensure that the curve maintains its full geometric dimension. When these conditions, often expressed using determinants or invariants, are satisfied, the conic has a well?defined center and shape.

Determinant of the Conic

The determinant formed from the coefficients of the quadratic and linear terms in the second degree equation plays a key role in determining whether the conic is degenerate or not. A non-zero value of this determinant ensures that the quadratic form, when completed to a square and translated to its center, maintains a non-zero constant term. This indicates a genuine conic section, whereas a zero determinant signals that the conic can degenerate.

Degenerate Conic

A degenerate conic occurs when the conditions that normally guarantee a full conic section fail—typically when the determinant associated with the quadratic form is zero. This situation leads to graphs that are not true curves but rather collapse into simpler figures such as a single point, a line, or a pair of lines. In this particular context, if the determinant equals zero, the graph does not represent a proper ellipse, hyperbola, or parabola.

Please give Ace some feedback