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 $A \neq 0, C=0$, and $E \neq 0$, then the equation is equivalent to one of the form

$$
(x-h)^2=4 p(y-k)
$$

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a cone. The general quadratic equation in two variables, Ax² + Cy² + Dx + Ey + F = 0, represents various types of conics (circles, ellipses, parabolas, and hyperbolas) depending on the values and relationships of the coefficients. Understanding the structure and classification of conic sections is crucial for determining the nature of the curve from the given equation.

Parabolas

A parabola is a specific type of conic section that can be defined as the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). Its standard equation, typically expressed in the form (x - h)² = 4p(y - k) or (y - k)² = 4p(x - h), provides information on the vertex, axis of symmetry, and the distance to the focus. Recognizing and rewriting an equation into this standard form is essential for studying the geometric properties of a parabola.

Completing the Square

Completing the square is an algebraic method used to transform a quadratic equation into a form that clearly reveals its geometric properties, such as the vertex form of a parabola. This technique involves rearranging and grouping terms related to one variable to form a perfect square trinomial, thereby simplifying the equation and making it easier to analyze and graph the conic section.

Coordinate Translation

Coordinate translation is a process used to shift the origin of the coordinate system to a new point (h, k) in order to simplify the equation of a conic section. By applying translations, the equation can often be expressed in a standard form, such as (x - h)² = 4p(y - k) for a parabola, which directly exhibits key features like the vertex and axis of symmetry. This concept is crucial when rewriting a general conic equation into a more informative and manageable form.

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