Answer
$r=\dfrac{8}{5+2\cos \theta}$
Work Step by Step
The conversion of polar coordinates and Cartesian coordinates are given as follows:
1. $r^2=x^2+y^2$ or, $r=\sqrt {x^2+y^2}$
2. $\tan \theta =\dfrac{y}{x}$ or,$ \theta =\tan^{-1} (\dfrac{y}{x})$
3. $x=r \cos \theta$ and 4. $y=r \sin \theta$
The equation of conic with eccentricity $e$ and directrix $d=x$ leads to focus can be written as:
$r=\dfrac{de}{1+e \cos \theta}$
Since, the focus is at the origin, so the directrix line must be half way of the vertex, Thus, the parabola with $e=0.4; d=4$
Thus, $r=\dfrac{1.6}{1+0.4\cos \theta}=\dfrac{8}{5+2\cos \theta}$
