$\frac{I_{1}}{I_{2}}=\frac{a^{2}}{b^{2}}=\beta \therefore \frac{a}{b}=\sqrt{\beta}$
Fringe visibility is given by
$V=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}=\frac{\left(a+b\right)^{2}-\left(a-b\right)^{2}}{\left(a+b\right)^{2}+\left(a-b\right)^{2}}$
$=\frac{4ab}{2\left(a^{2}+b^{2}\right)}=\frac{2\left(a/b\right)}{\left(\frac{a^{2}}{b^{2}}+1\right)}=\frac{2\sqrt{\beta}}{\beta+1}$