Suppose that A and C are not both 0 . Show that the set of...

Suppose that $A$ and $C$ are not both $0 .$ Show that the set of all $(x, y)$ satisfying $$ A x^{2}+B x+C y^{2}+D y+E=0 $$ is either a parabola, an ellipse, or an hyperbola (or possibly $\emptyset$ ). Hint: The case $C=0$ is essentially Problem $15,$ and the case $A=0$ is just a minor variant. Now consider separately the cases where $A$ and $B$ are both positive or negative, and where one is positive while the other is negative.

Suppose that $A$ and $C$ are not both $0 .$ Show that the set of all $(x, y)$ satisfying
$$
A x^{2}+B x+C y^{2}+D y+E=0
$$
is either a parabola, an ellipse, or an hyperbola (or possibly $\emptyset$ ). Hint:
The case $C=0$ is essentially Problem $15,$ and the case $A=0$ is just a minor variant. Now consider separately the cases where $A$ and $B$ are both positive or negative, and where one is positive while the other is negative.

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double?napped cone. The general quadratic equation in two variables represents these curves, and depending on the relationship between the coefficients, the curve can be a parabola, an ellipse (which includes the special case of a circle), or a hyperbola. Understanding these curves is fundamental in analytic geometry and many applications across mathematics and physics.

Completing the Square

Completing the square is an algebraic technique used to rewrite quadratic expressions in a form that makes the geometry of the conic clear by isolating squared binomials. By rewriting the quadratic equation into its standard form, one can readily identify the type of conic section represented, determine its center or vertex, and understand its orientation. This method is essential for transforming a general conic equation into a form that is easier to interpret and graph.

Classification of Conic Sections Using Coefficients

The classification of conic sections from their general quadratic equation depends on the coefficients of the squared terms. When both squared terms are present, one compares the coefficients: if they have the same sign, the conic is an ellipse (or a circle when the coefficients are equal), whereas if they have opposite signs, the conic is a hyperbola. If one of the squared terms is missing (i.e., its coefficient is zero), the equation represents a parabola. This method of classification allows for immediate recognition of the type of curve based on the algebraic form of the equation.

Degenerate Cases

Degenerate cases occur when the quadratic equation does not yield a typical conic section but instead results in simpler geometric objects such as a point, a line, or even an empty set. These arise when the process of completing the square leads to equations that have no real solutions or that reduce to trivial identities. Recognizing degenerate cases is important because it highlights the full range of possibilities encompassed by the general quadratic form, beyond the standard conic sections.

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