In a 3-phase induction motor the torque under running conditions will be maximum when:

Correct option is A

We know that -

$$T = \frac{3}{2 \pi N_s} \times \frac{s E_2^2 R_2}{R_2^2 + s^2 X_2^2}$$

$$\text{For maximum running torque:}$$

$$\frac{dT}{ds} = 0$$

$$K \left[ \frac{(R_2^2 + s^2 X_2^2)E_2^2 R_2 - s E_2^2 R_2 (2s X_2^2)}{R_2^2 + s^2 X_2^2} \right] = 0$$

$$(R_2^2 + s^2 X_2^2)E_2^2 R_2 = s E_2^2 R_2 (2s X_2^2)$$

$$R_2^2 + s^2 X_2^2 = 2s^2 X_2^2$$

$$R_2^2 = 2s^2 X_2^2 - s^2 X_2^2$$

$$R_2^2 = s^2 X_2^2$$

$$R_2 = s X_2$$​​​​​​​​​