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

$(a,b)$ is the mid-point of Line segment joining the points $A(10,-6)$ and $B(k,4)$

So,

$a=\frac{10+k}{2}$ and $b=\frac{-6+4}{2}=-1$

It is given that,

$a-2b=18$

Put $b=-1$

$a-2(-1)=18$

$a=18-2=16$

Now,

$a=\frac{10+k}{2}$

$16=\frac{10+k}{2}$

$k+10=32$

$k=22$

Distance between the points $(x_1,y_1)$ and $(x_2,y_2)$ is

$\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$

Distance between the points $(10,-6)$ and $(22,4)$ is

$\sqrt{(22-10{)}^2+(4+6{)}^2}$

$=\sqrt{244}$

$=2\sqrt{61}$