\(a_{2}=2 a, I_{2}=a_{2}^{2}=4 a^{2}\). Therefore \(I_{2}=4I_1\)

\(I_{r}=I_{1}+I_{2}+2 \sqrt{I_{1} I_{2}} \cos \phi\)

\(I_{r}=I_{1}+4 I_{1}+2 \sqrt{4 I_{1}^{2}} \cos \phi\)

\(\Rightarrow \quad I_{r}=5 I_{1}+4 I_{1} \cos \phi\)      ...... \((1)\)

Now, \(I_{\max }=\left(a_{1}+a_{2}\right)^{2}=(a+2 a)^{2}=9 a^{2}\)

\(I_{\max }=9 I_{1} \Rightarrow I_{1}=\frac{I_{\max }}{9}\)

Substituting in equation \((1)\)

\(I_{r}=\frac{5 I_{\max }}{9}+\frac{4 I_{\max }}{9} \cos \phi\)

\(I_{r}=\frac{I_{\max }}{9}[5+4 \cos \phi]\)

\(I_{r}=\frac{I_{\max }}{9}\left[5+8 \cos ^{2} \frac{\phi}{2}-4\right]\)

\(I_{r}=\frac{I_{\max }}{9}\left[1+8 \cos ^{2} \frac{\phi}{2}\right]\)