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 $A \neq C$, but $A C>0$, then the equation is equivalent to an equation of one of the forms $$ \begin{aligned} & \quad \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1 \quad \text { or } \quad \frac{(y-k)^2}{a^2}+\frac{(x-h)^2}{b^2}=1 \\ & a>b>0, \text { provided } A\left(4 A C F-C D^2-A E^2\right)<0 \end{aligned} $$

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 $A \neq C$, but $A C>0$, then the equation is equivalent to an equation of one of the forms

$$
\begin{aligned}
& \quad \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1 \quad \text { or } \quad \frac{(y-k)^2}{a^2}+\frac{(x-h)^2}{b^2}=1 \\
& a>b>0, \text { provided } A\left(4 A C F-C D^2-A E^2\right)<0
\end{aligned}
$$

Key Concepts

Classification Conditions for Conic Sections

The conditions such as A ? C, A·C > 0, and the sign of specific combinations of coefficients (for example, A(4ACF – CD² – AE²) < 0) are used to identify the type of conic section and ensure that the resulting equation indeed represents a non-degenerate ellipse. These conditions help in distinguishing ellipses from other conic sections by providing criteria based on the algebraic structure of the quadratic terms.

Standard Form of an Ellipse

The standard form of an ellipse is expressed as (x-h)²/a² + (y-k)²/b² = 1 (or vice versa), where (h, k) represents the center of the ellipse and a and b are the lengths of the semi-major and semi-minor axes. Converting a general quadratic equation into this form by completing the square simplifies the process of understanding the ellipse’s geometry and its orientation in the coordinate plane.

Quadratic Forms in Two Variables

A quadratic equation in two variables represents a conic section and is typically written in the form Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. In scenarios where the xy term (Bxy) is absent, the conic does not require rotation for its standardization, which simplifies the process of analyzing the conic's characteristics.

Completing the Square

Completing the square is a method used to convert a quadratic equation into a form that clearly reveals the center and dimensions of the conic. By rewriting the expression in terms of squared binomials, one can transition from the general quadratic form to the canonical form, making it easier to identify and analyze the conic section—especially in determining if it is an ellipse, parabola, or hyperbola.

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