LRC

Let the P.D. across the resistance be X, inductance be Y and capacitance be Z. Then $X^2$ + $Y^2$ = $8^2$ =64, $X^2$ + $Z^2$ = $5^2$ =25, and $X^2$ + $(Y-Z) ^2$ = $5^2$ =25. Solving we get $X^2$ =12. Therefore the power dissipated across the resistance is $\dfrac{12}{R}$ =12 Watts (since R=1 Ohm)

The unknown impedances are as follows:

$Z_L = 0 + j X_L \\ Z_C = 0 - j X_C$

Determine $X_L$ and $X_C$ such that $U_1$ has a magnitude of $8$ and $U_2$ has a magnitude of $5$ . I used a hill-climbing algorithm for this. Code and results are attached.

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import math
import random

E = complex(5.0,0.0)  # source voltage
R = 1.0  # resistance

###################################################

delta = 0.001  # mutation size

minres = 99999999.0 # initialize cumulative residual to large value 
                                     # eventually, this should become close to zero

XL = 5.0 * random.random()  # inductive reactance magnitude - random initialization
XC = 5.0 * random.random()  # capacitive reactance magnitude - random initialization

###################################################

for j in range(0,10**6):

    XLm = XL + delta * (-1.0 + 2.0 * random.random())   # mutate XL,XC
    XCm = XC + delta * (-1.0 + 2.0 * random.random())

    ZL = complex(0.0,XLm)  # calculate complex impedances
    ZC = complex(0.0,-XCm)

    I = E/(ZL + R + ZC)  # complex current
    Imag = abs(I)  # current magnitude

    U1 = I*(ZL+R)  # complex voltage U1
    U2 = I*(R+ZC)  # complex voltage U2

    res1 = math.fabs(abs(U1) - 8.0)  # difference between U1 magnitude and 8
    res2 = math.fabs(abs(U2) - 5.0)  # difference between U2 magnitude and 5

    res = res1 + res2  # form residual (should be zero)

    P = (Imag**2.0)*R  # power in resistor

    if res < minres:   # accept mutated XL,XC if cumulative residual gets smaller

        minres = res

        XL = XLm 
        XC = XCm

        P_store = P

###################################################

print "minimum residual"
print minres
print ""
print "XL - ohms"
print XL
print ""
print "XC - ohms"
print XC
print ""
print "resistor power"
print P_store

###################################################

# Results

#minimum residual
#2.59320195983e-06

#XL - ohms
#2.08166797428

#XC - ohms
#1.04083402865

#resistor power
#11.9999886584