Your first method is almost correct. You went wrong at the very last step, when you plugged in numbers. Hint: look at the units for every variable.
Your second method failed because you're not accounting for phase. \$V_C\$ is not in phase with \$V_e\$, \$V_A\$, or \$I\$, so you can't just add or subtract their magnitudes and get a physically meaningful value. In your first approach, you already found that:
$$|V_C| + |V_A| > |V_e|$$
This looks like it violates Kirchoff's Voltage Law, but it really doesn't. KVL says that:
$$v_c(t) + v_a(t) = v_e(t)$$
In terms of complex exponentials (aka cosines), that gives:
$$|V_C|e^{j(\omega t + \phi_C)} + |V_A|e^{j(\omega t + \phi_A)} = |V_e|e^{j(\omega t + \phi_e)}$$
which in turn means that:
$$|V_C|e^{j\phi_C} + |V_A|e^{j\phi_A} = |V_e|e^{j\phi_e}$$
or, in phasor notation:
$$V_C \angle \phi_C + V_A \angle \phi_A = V_e \angle \phi_e$$
Try again with the phases included and you should be able to get the right answer.
EDIT: The equation you've come up with is:
$$100 \mathrm V \angle \phi_e = V_C \angle {- \frac \pi 2} + 50 \mathrm V \angle 0$$
This is one equation in two unknowns. You need another equation. There are two options:
$$\phi_C = \tan^{-1} \frac{-V_C}{V_A} = \tan^{-1} \frac {-V_C}{50 \mathrm V}$$
$$|V_C|^2 = |V_e|^2 - |V_A|^2 = (100 \mathrm V)^2 - (50 \mathrm V)^2$$
Inverse tangents are awkward to deal with, since they tend to confuse computer algebra systems. The second equation is much easier, can be solved by itself, and leads directly to the answer you want.
Best Answer
So yo had the denominator wrong as it should have been -7 + j17 instead of -6 + j17. I don't quite see the point of converting it back and forth between Cartesian and polar forms; you can calculate it directly will less work; and obviously (-3*5)+4+4=-7.
Your error however is a lot more trivial than that, you carried over a 4 to a 5: