Let and let be any real parameter

MATHEMATICS · Multi-part · CBSE · Mathematics · Conic Sections · Conic Sections

Problem

Let

a>0

and let

t

be any real parameter. Define the point

P(at^{2},2at)

in the

xy

-plane. (i) Show, by direct substitution, that

P(at^{2},2at)

satisfies the equation

y^{2}=4ax

. (ii) Compute the distance of

P

from the focus

F(a,0)

and the perpendicular distance of

P

from the directrix

x=-a

, and prove that these two distances are equal for every real

t

. (iii) Explain briefly how the result in part (ii) confirms that the set of points

\{(at^{2},2at)\,|\,t\in\mathbb{R}\}

exhausts the entire parabola

y^{2}=4ax