Question

Find the equation of the parabola with focus $(2,0)$ and directrix $x=-2$.Also find the length of the latus rectum.

Answer 1

The focus of the parabola is $F(2,0)$ and its directrix is the line $x=-2$ i.e., $x+2=0$

Let $P(x,y)$ be any point in the plane of directrix and focus, and $MP$ be the perpendicular distance from $P$ to the directrix,then $P$ lies on parabola iff $FP=MP$

$\Rightarrow \sqrt{{(x-2)}^2+{(y-0)}^2}=\frac{|x+2|}{1}$

$\Rightarrow x^2-4x+4+y^2=x^2+4x+4$

$\Rightarrow -4x+y^2=4x$

$\Rightarrow y^2=8x$ which is the required equation of the parabola.

Comparing it with $y^2=4ax$ we get $4a=8\Rightarrow a=\frac{8}{4}=2$

$\therefore$Length of the latus rectum$=4a=4\times 2=8$