$v _{0}=\alpha t _{1}$ and $0= v _{0}-\beta t _{2} \Rightarrow v _{0}=\beta t _{2}$
$t _{1}+ t _{2}= t$
$v _{0}\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)= t$
$\Rightarrow v _{0}=\frac{\alpha \beta t }{\alpha+\beta}$
Distance $=$ area of $v – t$ graph
$=\frac{1}{2} \times t \times v _{0}=\frac{1}{2} \times t \times \frac{\alpha \beta t }{\alpha+\beta}=\frac{\alpha \beta t ^{2}}{2(\alpha+\beta)}$