The Parabola

Share this:

Parabola definition: locus of P such that PS=PL.

Cartesian Equations:

(x-a)^{2}+y^{2}=(x+a)^{2}
x^{2}-2ax+a^{2}+y^{2} = x^{2}+2ax+a^{2}
y^{2} = 4ax

with gradient:

2y\dfrac{\textnormal{d}y}{\textnormal{d}x}=4a\ \implies\ \dfrac{\textnormal{d}y}{\textnormal{d}x}=\dfrac{2a}{y}

Parametric Equation:

x=at^{2},\ y=2at

with gradient:

\dfrac{\textnormal{d}x}{\textnormal{d}t}=2at,\ \dfrac{\textnormal{d}y}{\textnormal{d}t}=2a\ \implies\ \dfrac{\textnormal{d}y}{\textnormal{d}x}=\dfrac{1}{t}

tangent:

y-2at=\dfrac{1}{t}(x-at^{2})\ \implies ty=x+at^{2}

normal:

y-2at=-t(x-at^{2})\ \implies y+tx=2at+at^{3}

Leave a Comment

Your email address will not be published. Required fields are marked *

nine + 19 =