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 .
Show that if $A=C \neq 0$, then the equation is equivalent to one of the form

$$
(x-h)^2+(y-k)^2=q .
$$

What must be true of $q$ if the equation is to have a nonempty graph in $\Omega^2$ ?

Key Concepts

Non-Negativity of the Radius Squared (q)

For a circle with equation (x - h)² + (y - k)² = q to represent a nonempty graph in the plane, the value of q must be nonnegative. A positive q corresponds to a circle with a positive radius, while q equal to zero results in a degenerate circle, which is essentially a single point. If q were negative, there would be no real solutions, meaning the graph would be empty.

Degenerate Cases in Conic Sections

Degenerate conic sections occur when the parameters reduce the conic to a simpler set such as a point, a line, or even an empty set. In the context of a circle, when the value of q equals zero, the circle collapses into a single point, and if q is less than zero, the equation does not correspond to any real geometric figure.

Completing the Square

Completing the square is an algebraic technique used to transform a quadratic expression into a perfect square plus or minus a constant. This method is pivotal in rewriting quadratic equations into a form that clearly reveals the geometric properties of the conic section, such as its center and radius when dealing with circles.

Standard Form of a Circle

The standard form of the equation of a circle is given by (x - h)² + (y - k)² = r², where (h, k) represents the center and r is the radius. This formulation makes it easy to identify the essential characteristics of the circle and to analyze its geometry.

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