The voltage required across the lamp is 100V and at this voltage the current through both bulb and capacitor will be I=0.5 A to dissipate 50W in the bulb.
First we need the voltages across bulb and capacitor to combine to give 200V. With the current through each being in phase then the voltage V across the capacitor will be 90° out of phase with that across the bulb (which I’m assuming is incandescent and therefore purely resistive). So we need to use Pythagoras to combine them:
1002+V2=2002
and so V=1003–√ .
Now the relationship between AC voltage and current for a capacitor is
I=2πfCV
where frequency f is 50Hz in this case. Solving for C and inserting the values for I and V calculated above we get:
C=I2πfV
=0.510000π3–√
=50π3–√μF
This is approximately 9.19μ F .