Induction Motor Question
I shall use the following notations : rated speed is \$N_r\$, synchronous speed \$N_s\$, torque \$T\$, rated torque \$T_r\$, operating torque \$T_{op} = \frac{2}{3} \times T_r\$.

When you write "Assume a linear τ - ω relationship near synchronous speed", this means:
$$T = K (N_s - N)$$
for any speed \$N\$ near synchronous speed and corresponding torque \$T\$. This is also true for rated torque and speed:
$$T_r = K (N_s - N_r)$$
and for operating torque and associated speed:
$$T_{op} = K (N_s - N_{op})$$
The two last equations yield:
$$N_s - N_{op} = \frac{T_{op}}{K} = \frac{T_{op}}{T_r}(N_s - N_r)$$

which should give you the answer.