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

The correct option is C.$22$

Let the point $P(x,y)=(a,b)$ be the mid point of the line segment $AB$ joining the points $A(x_1,y_1)=(10,6)\&B(x_2,y_2)=(k,-4)$ in the ratio $m:n=1:1$

Then, by the section formula,

$x=\frac{m{x}_2+n{x}_1}{m+n}=\frac{x_2+x_1}{2}$

$\Rightarrow a=\frac{10+k}{2}\Rightarrow k=2a-10$ ...(i)

$y=\frac{n{y}_1+m{y}_2}{m+n}=\frac{y_2+y_1}{2}$

$\Rightarrow b=\frac{6-4}{2}=1$ ...(ii)

But it is given that $a+2b=18\Rightarrow a=18-2b=18-2\times 1=16$ [From (i) and (ii)]

Substituting $a=16$ in (i) we get $k=2\times 16-10=22$

$\therefore a=16,k=22$