Is my thinking right for the simple harmonic circuit
If you drill down through the equation for the impedance of the circuit and equate the imaginary terms to zero you find that the frequency where the phase angle is purely resistive is: -

$$\omega = \sqrt{\dfrac{1}{LC} -\dfrac{R^2}{L^2}}$$

Then, if you square both sides and merge \$\omega\$ with the L and C terms to produce impedance terms you get: -

$$1 = \dfrac{X_C}{X_L} - \dfrac{R^2}{X_L^2}$$

$$\dfrac{X_C}{X_L} = 1 + \dfrac{R^2}{X_L^2}$$

$$X_C = X_L + \dfrac{R^2}{X_L}$$

So, 

$$X_C = 12 + \dfrac{1^2}{12^2} = 12.0833\text{ ohms}$$