<s>[INST] Solving $ \frac{1}{ a} = \ \frac{1}{ \ \sqrt{b}} \ +\ \frac{1}{ \ \sqrt{c}} $ with additional conditions How to solve this equation 
$$ \frac{1}{a}= \  \frac{1}{ \ \sqrt{b}} \ +\  \frac{1}{ \ \sqrt{c}} $$
where $$ b =  \sqrt{ (x-a/2)^2 + y^2 + z^2 )}$$
&   $$ c =  \sqrt{ (x+a/2)^2 + y^2 + z^2 )}$$
We have to get an equation in x y z and a!
I have tried to rationalize and do squaring but it becomes cumbersome.
Answer given is  $y^2+z^2=15/4$
I get this by putting x=0.
 [/INST] Partial answer: You need a simplification, not a solution.You already have an implicit equation in x, y, z and a ! The surface is symmetric about x = 0 or y-z plane ( change of sign before x makes no change). One needs to study (x,y) intersection only, as it is formed by rotation about x-axis ( due to appearance of  $ y^2+z^2$ ).
EDIT: From the graphic, plane x=0 does not cut the surface along any real line of intersection.
It appears like, may be two (higher order?) spheres, centered at $ (\pm a/2,0,0)$ as seen in following Mathematica 3D plot. $a$ is taken as unity.

EDIT1: 
Patience with algebraic simplification ( which is the solution of problem of seeing what you are dealing with!)  is all that is needed.
$\sqrt{bc}/ ( \sqrt b + \sqrt c) = a $
$ (b c /a^2)^2 + (b+c)^2 - 2 b c (b+c)/ a^2 = b^2 c^2  $
Let $ ( b + c) / ( b c) = u $
$ u^2 + -2 u /a^2 + (1/a^4-1) $
$ u = 1/b + 1/c = 1/a^2 \pm 1 = 1/d $
So due to d there are two surfaces 
$ 2 (x^2 + a^2/4 + y^2 + z^2) +2 b c = 1/d^2(  (x^2 + a^2/4 + y^2 + z^2)^2 -( 2 a x)^2 $
which shows the two ordinary displaced fourth order  spheres.
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