Question

If (a,b) is the mid-point of the line segment joining the points A (10, $-$6) , B (k,4) and a$-$2b = 18 , find the value of k and the distance AB.

Answer 1

It is given that A(10, −6) and B(k, 4).
Suppose (a, b) be midpoint of AB. Then,
$$\begin{gathered} a=\frac{10+k}{2},b=\frac{-6+4}{2}=\frac{-2}{2}=-1 \\ Now,a-2b=18 \\ \Rightarrow a=18-2=16 \\ Therefore, \\ 16\times 2=10+k \\ \Rightarrow k=22 \\ Further, \\ AB=\sqrt{{\left(22-10\right)}^2+{\left(4+6\right)}^2} \\ =\sqrt{144+100} \\ =2\sqrt{61} \end{gathered}$$