As it is given (a, b) is the mid-point of the line segment A($10,-6$) and B(k,$4$)
Therefore,
(a, b) $=(10+k/2,-6+4/2)$
a $=(10+k)/2$ and b $=-1$
$2a=10+k$
$K=2a–10$
Given that, $a–2b=18$
By Using $b=-1$ in the above relation we get,
$a-2(-1)=18$
A$=18-2=16$
So,
k $=2(16)-10=32-10=22$
Hence, $AB=\sqrt{\left[ {{(22-10)}^{2}}+{{(4+6)}^{2}} \right]}$
$=\sqrt{\left[ {{(12)}^{2}}+{{(10)}^{2}} \right]}$
$=\sqrt{\left[ 144+100 \right]}$
$=2\sqrt{61}$