As initially $q$ is at infinite distance from $Q$, thus $P \cdot E_i=0$

For case 1: the closest distance given is $r$ and velocity of $q$ is $v$ initially.

Applying conservation of energy, $P . E_i+K . E_i=P . E_f+K . E_f$

$0 +\frac{1}{2} mv ^2=\frac{ KQq }{ r }+0$

$\Rightarrow \frac{1}{2} mv ^2 =\frac{ KQq }{ r } \quad \ldots \ldots \ldots \ldots \ldots . . .(1)$

For case $1$: Let the closest distance be $r^{\prime}$ and velocity of $q$ is $2 v$ initially.

Applying conservation of energy, $P \cdot E_i+K \cdot E_i=P \cdot E_f+K \cdot E_f$

$+\frac{1}{2} m (2 v )^2=\frac{ KQq }{ r ^{\prime}}+0$

$\Rightarrow \frac{1}{2} m (2 v )^2=\frac{ KQq }{ r ^{\prime}} \quad \ldots \ldots \ldots \ldots . . .(2)$

Solving $(1)$ and $(2)$, $r^{\prime}=\frac{r}{4}$