Answer
The two phasors have the same length when $~~f_d = 159~Hz$
Work Step by Step
We can write an expression for $V_R$:
$V_R= I_R~R$
We can write an expression for $V_C$:
$V_C= I_C~X_C$
We can equate the two expressions to find $f_d$:
$V_R = V_C$
$I_R~R = I_C~X_C$
$R = X_C$
$R = \frac{1}{\omega_d~C}$
$R = \frac{1}{2\pi~f_d~C}$
$f_d = \frac{1}{2\pi~R~C}$
$f_d = \frac{1}{(2\pi)~(50.0~\Omega)~(20.0\times 10^{-6}~F)}$
$f_d = 159~Hz$
The two phasors have the same length when $~~f_d = 159~Hz$
