Question

the heater is designed to operate with power of 1000 watts in a 100 volts line it is connected in a combination with resistance R to 100V mains
as shown in figure what should be the value of our society Thambi operate with the power of 62.5 watts

Answer 1

You know the initial watt and volt, then find the resistance of heater using V²R=W

Then you are given W=62.5 and you know the resistance of heater. From here find the voltage under which the heater works.

Now you know the voltage across the parallel combination

Now you have 2 ways:
1) Find current in the heater (you know the resistance and voltage), find net current in circuit and you know that current is inversely distributed in parallel combination, so get relations in R and get the answer

2) Get equation regarding voltage is same in R and heater.

Answer 2

When connecting with heater at a power of 1000W in a 100V line, the resistance will trigger according to the connections.

This usually giving power of 62.5 W and used to calculate the resonance power of 10 Ohm and R. The heater is now giving a power of 62.5W.