When $0.1\, N NaOH$ is used,
$\underset{\text{(For HCl)}}{N_{1} V_{1}}= \underset{\text{(For NaOH)}}{N_{2} V_{2}}$
$0.2\, N \times V_{1}=50 \times 0.1\, N$
$V_{1}=\frac{50 \times 0.1}{0.2}=25\, cm ^{3}$
When $0.5\, N KOH$ is used,
$\underset{\text{(For remaining HCl)}}{N_{1} V_{1}}=\underset{\text{(For KOH)}}{N_{3} V_{3}}$
$0.2\, N \times 25 =0.5\, N \times V_{3}$
$V_{3} =\frac{0.2 \times 25}{0.5}$
$=10\, cm ^{3}$