Question

Find the equation of the parabola with focus $(8,0)$ and directrix $x=-8$.Also find the length of the latus rectum.

Answer 1

The focus of the parabola is $F(8,0)$ and its directrix is the line $x=-8$ i.e., $x+8=0$

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 $FP=MP$

$\Rightarrow \sqrt{{(x-8)}^2+{(y-0)}^2}=\frac{|x+8|}{1}$

$\Rightarrow x^2-16x+64+y^2=x^2+16x+64$

$\Rightarrow -16x+y^2=16x$

$\Rightarrow y^2=32x$ which is the required equation of the parabola.

Comparing it with $y^2=4ax$ we get $4a=32\Rightarrow a=\frac{32}{4}=8$

$\therefore$Length of the latus rectum$=4a=4\times 8=32$