$t _{1}+ t _{2}= t$

$v _{0}\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)= t$

$\Rightarrow v _{0}=\frac{\alpha \beta t }{\alpha+\beta}$

Distance $=$ area of $v - t$ graph

$=\frac{1}{2} \times t \times v _{0}=\frac{1}{2} \times t \times \frac{\alpha \beta t }{\alpha+\beta}=\frac{\alpha \beta t ^{2}}{2(\alpha+\beta)}$