The Parabola - jped Maths

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:

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

Cartesian Equations:

Parametric Equation:

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