Find eccentricity of ellipse

Question

This problem bothers me a bit: Find eccentricity of ellipse if distance between its foci is arithmetical average of length of semi major and semi minor axis. Well I know that e=c/a and c^2 = a^2 - b^2. 2c=a+b, should I just plugin inside e=c/a for c? Is there something I am missing here?

Answer 1

$4a^2-4b^2=4c^2= (a+b)^2=a^2+2ab+b^2.$ So $3a^2-5b^2-2ab=0.$ Dividing thru by $a^2$ and putting $b/a=e,$ we have $3-5e^2-2e=0.$ With one positive root, $e=3/5.$