Q.Define parabola and obtain the standard form of the parabola $y^2=4ax, \ (a>0)$.

A parabola is the locus of points equidistant from a fixed point (focus) and a fixed line (directrix); placing the focus at $(a,0)$ and directrix at $x=-a$ and equating the two distances gives $y^2=4ax$.

Definition: A parabola is the locus of a point which moves in a plane such that its distance from a fixed point (the focus) is always equal to its perpendicular distance from a fixed straight line (the directrix), the focus not lying on the directrix.

Derivation: Take the focus $S=(a,0)$, $a>0$, and the directrix as the line $x=-a$. Let $P(x,y)$ be any point on the parabola.

Distance from $P$ to the focus: $SP=\sqrt{(x-a)^2+y^2}$

Distance from $P$ to the directrix: $x+a$

By the defining property, $SP$ equals this distance:

$\sqrt{(x-a)^2+y^2}=x+a$

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