Show that the graph of an equation of the form A x^2+C y^2+D x+E y+F=0 A ≠0, C ≠0 where A and

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Show that the graph of an equation of the form A x^2+C y^2+D...

Show that the graph of an equation of the form $$ A x^{2}+C y^{2}+D x+E y+F=0 \quad A \neq 0, C \neq 0 $$ where $A$ and $C$ are of the same sign, (a) is an ellipse if $\frac{D^{2}}{4 A}+\frac{E^{2}}{4 C}-F$ is the same sign as $A$. (b) is a point if $\frac{D^{2}}{4 A}+\frac{E^{2}}{4 C}-F=0$. (c) contains no points if $\frac{D^{2}}{4 A}+\frac{E^{2}}{4 C}-F$ is of opposite sign to $A$.

Show that the graph of an equation of the form
$$
A x^{2}+C y^{2}+D x+E y+F=0 \quad A \neq 0, C \neq 0
$$
where $A$ and $C$ are of the same sign,
(a) is an ellipse if $\frac{D^{2}}{4 A}+\frac{E^{2}}{4 C}-F$ is the same sign as $A$.
(b) is a point if $\frac{D^{2}}{4 A}+\frac{E^{2}}{4 C}-F=0$.
(c) contains no points if $\frac{D^{2}}{4 A}+\frac{E^{2}}{4 C}-F$ is of opposite sign to $A$.

Key Concepts

Degenerate Conic Sections

Degenerate conic sections occur when the quadratic equation does not represent a typical curve but instead simplifies to a lower-dimensional set, such as a single point, a line, or even an empty set. The conditions extracted from the constant term (after completing the square) determine whether the quadratic equation corresponds to a proper conic section or a degenerate case.

Ellipses

An ellipse is a conic section defined by the set of points for which the sum of the distances from two fixed points (the foci) is constant. When the quadratic equation is arranged into standard form through completing the square, the coefficients and the constant term reveal important information about the size, orientation, and existence of the ellipse.

Conic Sections

Conic sections are curves obtained from the intersection of a plane with a double-napped cone. They include ellipses, parabolas, and hyperbolas, and their algebraic representations are given by quadratic equations in two variables. The study of these curves involves determining the type of conic (ellipse, circle, hyperbola, etc.) based on the coefficients of the equation.

Completing the Square

Completing the square is an algebraic method used to transform quadratic expressions into a perfect square plus a constant term. This technique is crucial for rewriting the quadratic equation in a standard form, which makes it easier to identify the center and nature of the conic section, such as distinguishing between a true ellipse and its degenerate cases.

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