\(2 R_A=R_B\)

\(2 \lambda_1 N_1=\lambda_2 N_2 \quad \dots (i)\)

Radio-activity is same after say time \(t\)

\(\lambda_1 N_1 e^{-\lambda_1 t}=\lambda_2 N_2 e^{-\lambda_2 t} \quad \dots (ii)\)

Dividing \((i)\) by \((ii)\)

\(2 e^{\lambda_1 t}=e^{\lambda_2 t}\)

\(2=e^{\left(\lambda_2-\lambda_1\right) t}\)

Taking In on both sides

\(0.693=\left(\lambda_2-\lambda_1\right) t\)

\(1=\left(\frac{1}{T_2}-\frac{1}{T_1}\right) t\)

\(\frac{T_2 T_1}{T_1-T_2}=t\)