Mathematics

Problem

Let the focus be

F(a,0)

with

a>0

and the directrix be the line

x=-a

. A variable point

P(x,y)

moves in the plane. a. (i) State the geometric definition of a parabola with respect to

F

and the given directrix. b. (ii) Using the condition

PF=\text{distance of }P\text{ to the directrix}

, derive the algebraic relation satisfied by the coordinates

(x,y)

P

and hence obtain the standard equation of the curve. c. (iii) Conversely, starting from the equation

y^{2}=4ax

, prove that any point

(x,y)

on this curve is equidistant from

F

and the directrix, thereby confirming the locus interpretation.