Question 446169
Without loss of generality, let h > 0, let (h, 0) be the center of the ellipse, and let the point (h + c, 0) be the other focus. Since the (absolute) distance of the center of the ellipse from either foci is c = a*e (where a = semi-major axis, and e = eccentricity), this implies that h = c.
Also, h - a*e = c - a*e = 0 by hypothesis, hence, c = a/2.

Now using the directrix condition, we get h + a/e = 4, or 
h + a/(1/2) = h + 2a = 4.  After using the fact that h = c, we get c + 2a = 4.

Hence by substitution, a/2 + 2a = 4, 
or {{{(5a)/2 = 4}}}, or
{{{a = 8/5}}}, the length of the semi-major axis.