Coefficient of coupling in Resonance circuit
This question is hanging around so I'm going to give some more help. No need for any upvotes.

As Marla said "For applied input current and voltage to be in phase, the reactive components have to cancel out to zero".

Do you understand the implications of this - do you see that the two mutually coupled inductors have to provide a reactance of +j12 so that numerically it becomes a series tuned circuit i.e. the capacitive -j12 is cancelled by the +j12 of the two series coils?

This is the important first step and if you don't understand why this produces a current in phase with the applied voltage then you need to do some back ground work.

If you follow this then the problem boils down to how you make the two coupled coils have a net impedance of +j12. Consider this: -



Do you see that ONLY when the inductor dots are "aiding" do you get an inductance increase over the basic L1 + L2? If you don't understand mutually coupled coils then you will have to do some research.

So, this tells you where to place the dot on coil PQ. 

Somewhere between zero coupling and 100% coupling is where you want to be. You are aiming for +j12 but you only have +j8 and +j2 - this means the extra +j2 comes from the mutual coupling. Using the formula above clearly M is equivalent to +j1.

If you did some research on the web you would find this formula: -



Because reactance is proportional to inductance you can substitute the reactances in that formula and derive K thus: -

\$K = \dfrac{M}{\sqrt{L1L2}} = \dfrac{1}{\sqrt{16}}\$ = 0.25