Thanks Jugurtha! so is the final answer: (Zb + Za)/(Zc + Zd) ?
When they ask you about the conditions to get a null current through ZE, something like (Zb+Za)/(Zc+Zd) doesn't mean anything .. It's not an equation or something ..
You got asked what are the conditions to get a null current through ZE, you START from this, from what you have been asked and take it as granted, as your starting point, then you "follow" the consequences of a null current through ZE which are:
*ZA and ZB are juiced by the same current.
*ZC and ZD are juiced by the same current.
VY=VZD=VX=VZA
VIN=VX+VZB=VY+VZC , with VX = VY, so they cancel each other we'll have:
VZB=VZC
What does this mean, we write that in currents going through them:
VZB=ZB*(I-Ic)
VZC=ZC*Ic
VZB=VZC ==> ZB(I-Ic)=ZC*Ic .... (1)
VX=VY , written another way ZA(I-Ic)=ZD*Ic ..... (2)
We divide (1) by (2) to get rid of currents to find the CONDITION to have a NULL current on ZE.
So (1)/(2) gives : ZB/ZA=ZC/ZD=(ZB+ZC)/(ZA+ZD) or written in another way, ZB*ZD=ZC*ZA.
That's the condition one the impedances ZA, ZB, ZC and ZD to get a null current on ZE.
So in your circuit, if you chose for example ZB=10 ohms, ZD=2 ohms, ZC=1 ohms, ZA=20 ohms, you'll get a null current on ZE.
Have a look on Wheatstone bridge, but no matter what, in this kind of problems, you are given a constraint, and you kind of "backward" engineer or reverse stuff to get to *satisfy* this constraint.
(In Control Systems for example, you are given desired characteristics of a system (overshoot, damping, pulsation -for a specific sampling time for example-), and you -backward find- its discrete transfer function, and backward find the propper corrector to do that, R-S-T or a special case Pole Placements, etc.. You do the reverse path, you *start from the end* and find the setting to get the desired output)
Good luck,