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Hey,

How do you derive the equations of the parabola from the general equation of a Conic Section?

General Equation of a Conic Section,

[tex]

{{{A}{{x}^{2}}}+{{B}{x}{y}}+{{C}{{y}^{2}}}+{{D}{x}}+{{E}{y}}+{F}} = {0}

[/tex]

Where (for a parabola),

[tex]

{{\{}A,B,C,D,E,F{\}}}{{\,}{\,}{\,}}{\in}{{\,}{\,}{\,}}{\mathbb{R}}

[/tex]

[tex]

{{\{}A, C{\}}}{{\,}{\,}{\,}}{\neq}{{\,}{\,}{\,}}{0}

[/tex]

[tex]

{{{B}^{2}}-{{4}{A}{C}}} = {0}

[/tex]

From the general equation of a Conic Section,

[tex]

{{{A}{{x}^{2}}}+{{B}{x}{y}}+{{C}{{y}^{2}}}+{{D}{x}}+{{E}{y}}+{F}} = {0}

[/tex]

How do I derive the following formulas for a parabola:

General Form for a Parabola,

[tex]

{{{{\left(}{{{H}{x}}+{{I}{y}}}{\right)}}^{2}}+{{J}{x}}+{{K}{y}}+{L}} = {0}

[/tex]

Where,

[tex]

{{{I}^{2}}-{{4}{H}{J}}} = {0}

[/tex]

Analytic Geometry Equations,

Vertical Axis of Symmetry

[tex]

{{{\left(}x-h{\right)}}^{2}} = {{{4}{p}}{{{\left(}y-k}{\right)}}}

[/tex]

[tex]

{y} = {{{a}{{x}^{2}}}+{{b}{x}}+{c}}

[/tex]

Horiztonal Axis of Symmetry

[tex]

{{{\left(}y-k{\right)}}^{2}} = {{{4}{p}}{{{\left(}x-h{\right)}}}}

[/tex]

[tex]

{x} = {{{d}{{y}^{2}}}+{{e}{y}}+{f}}

[/tex]

I read that every parabola is a combination of transformations of the parabola, [tex]{{y}={{x}^{2}}}[/tex]; but I'm not quite sure how that helps.

Thanks,

-PFStudent

## Homework Statement

How do you derive the equations of the parabola from the general equation of a Conic Section?

## Homework Equations

General Equation of a Conic Section,

[tex]

{{{A}{{x}^{2}}}+{{B}{x}{y}}+{{C}{{y}^{2}}}+{{D}{x}}+{{E}{y}}+{F}} = {0}

[/tex]

Where (for a parabola),

[tex]

{{\{}A,B,C,D,E,F{\}}}{{\,}{\,}{\,}}{\in}{{\,}{\,}{\,}}{\mathbb{R}}

[/tex]

[tex]

{{\{}A, C{\}}}{{\,}{\,}{\,}}{\neq}{{\,}{\,}{\,}}{0}

[/tex]

[tex]

{{{B}^{2}}-{{4}{A}{C}}} = {0}

[/tex]

## The Attempt at a Solution

From the general equation of a Conic Section,

[tex]

{{{A}{{x}^{2}}}+{{B}{x}{y}}+{{C}{{y}^{2}}}+{{D}{x}}+{{E}{y}}+{F}} = {0}

[/tex]

How do I derive the following formulas for a parabola:

General Form for a Parabola,

[tex]

{{{{\left(}{{{H}{x}}+{{I}{y}}}{\right)}}^{2}}+{{J}{x}}+{{K}{y}}+{L}} = {0}

[/tex]

Where,

[tex]

{{{I}^{2}}-{{4}{H}{J}}} = {0}

[/tex]

Analytic Geometry Equations,

Vertical Axis of Symmetry

[tex]

{{{\left(}x-h{\right)}}^{2}} = {{{4}{p}}{{{\left(}y-k}{\right)}}}

[/tex]

[tex]

{y} = {{{a}{{x}^{2}}}+{{b}{x}}+{c}}

[/tex]

Horiztonal Axis of Symmetry

[tex]

{{{\left(}y-k{\right)}}^{2}} = {{{4}{p}}{{{\left(}x-h{\right)}}}}

[/tex]

[tex]

{x} = {{{d}{{y}^{2}}}+{{e}{y}}+{f}}

[/tex]

I read that every parabola is a combination of transformations of the parabola, [tex]{{y}={{x}^{2}}}[/tex]; but I'm not quite sure how that helps.

Thanks,

-PFStudent

Last edited: