Question

a carnot cycle operate as a heat engine between two bodies of equal heat capacities until their temperature becomes equal if initial temperature of bodies s T1 ,T2 then t their common temperature

Answer 1

$$1-\frac{\delta Q_C}{\delta Q_H}=1-\frac{T_C}{T_H} (1)$$

where

$$\delta Q_C=C_pdT_c$$

$$\delta Q_H=C_pdT_H$$

Integrating that condition through the whole process

$$T=\sqrt{T_1T_2}$$

Answer:

$$T=\sqrt{T_1T_2}$$