What Is the General Equation of a Parabola and Its Derivation?

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The general equation of a parabola is represented as Ax² + Bxy + Cy² + Dx + Ey + F = 0. This equation is derived from the conic sections, which include parabolas, ellipses, circles, and hyperbolas, depending on the values of A, B, and C. Specifically, for the equation to represent a parabola, the condition B² = 4AC must be satisfied. Understanding this relationship is crucial for identifying the type of conic section represented by the equation. The discussion emphasizes the importance of these parameters in determining the geometric shape described by the equation.
DJ24
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I read a Wikipedia article titled "Parabola" that listed an equation of the form:

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

How is this derived? or where did it come from?
 
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This probably should be under the General Math section of the forums. The equation that you posted is the general quadratic equation and depending of the values of A, B, and C it can either produce parabolas, ellipses, circles, or hyperbolas (excepting a few degenerate cases). If it is a parabola then B2 = 4(A)(C).
 

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