Electronic – Why three-phase power? Why not a higher number of phases

The correct answer is obtained by making the absolute value of the imaginary part of the total equivalent impedance as small as possible (explanation follows below). If possible, make it zero, as this will yield the maximum possible power factor, which is one.

In your case, assuming you did all the delta-wye and wye-delta conversions correctly (I didn't check that), then the correct answer is given by setting the imaginary part of \$Z_3\$ to zero, that is: $$ 25⋅10^6 C^2 − 2⋅10^4 C + 3 = 0 $$ which, as you mention in the comments to another reply, yields two valid solutions, \$C=6·10^{-4} F\$ and \$C=2·10^{-4} F\$. If this equation had no positive roots, then you would have to look for a minimum of its absolute value.

Explanation

When people talk about the power factor of a given device without any reference to the voltage and current applied to it, they are implicitly thinking about connecting it to an ideal voltage source \$V\$. The expression for the apparent power consumed by the device in this simple circuit is \$S=V I^*\$, and the real power \$P\$ is just the real part of \$S\$. The power factor is defined as: $$ \cos\phi = \frac{P}{|S|} $$ By Ohm's law, \$V=IZ\$, therefore \$S = V I^*= |I|^2 Z = |I|^2 R + j|I|^2 X\$. So the power factor can be written as: $$ \cos\phi = \frac{|I|^2 R}{|I|^2 \sqrt{R^2+X^2}} = \frac{R}{\sqrt{R^2+X^2}} $$ As you can see, neither the voltage nor the current appear in the final expression. Also, in order to maximize the power factor (the original question), it becomes obvious that you need to minimize \$|X|\$. To be more precise, what maximizes the power factor is the minimization of the ratio \$|X|/R\$, because if we divide both numerator and denominator by R, we obtain: $$ \cos\phi = \frac{R}{\sqrt{R^2+X^2}} = \frac{1}{\sqrt{1+(\frac{X}{R})^2}} $$

Why not three lines all in the same phase?

1. Because then there is no return path.
2. Because single phase has no "rotation". Three phase makes it very simple to make a rotating motor with phase sequence determining the direction of rotation. Swap two phases and the direction is reversed.

Is there less loss when the phases of the three lines are all different?

1. Three phase power distribution requires less copper or aluminium for transferring the same amount of power as compared to single phase power.
2. The size of a three phase motor is smaller than that of a single phase motor of the same rating.
3. Three phase motors are self starting as they can produce a rotating magnetic field. The single phase motor requires a special starting winding as it produces only a pulsating magnetic field.
4. In single phase motors, the power transferred in motors is a function of the instantaneous power which is constantly varying. In three-phase the instantaneous power is constant.
5. Single phase motors are more prone to vibrations. In three phase motors, however, the power transferred is uniform through out the cycle and hence vibrations are greatly reduced.
6. Three phase motors have better power factor regulation.
7. Three phase enables efficient DC rectification with low ripple.

Figure 1. Resultant DC from three-phase rectifier.

8. Generators also benefit by presenting a constant mechanical load through the full revolution, thus maximising power and also minimising vibration.