- #1
PFStudent
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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: