Q.Find the coordinates of the point which divides the line segment joining $A(1, 2)$ and $B(7, 8)$ internally in the ratio $1:2$.

This is a direct application of the section formula for internal division, with $m:n=1:2$, $A(1,2)$ as the first point and $B(7,8)$ as the second.

The section formula for a point $P(x,y)$ dividing $A(x_1,y_1)$ and $B(x_2,y_2)$ internally in the ratio $m:n$ is

Here $A(x_1,y_1)=(1,2)$, $B(x_2,y_2)=(7,8)$, and the ratio is $m:n=1:2$. Substituting:

$y=\dfrac{(1)(8)+(2)(2)}{1+2}=\dfrac{8+4}{3}=\dfrac{12}{3}=4$ …

The single most common error is swapping which point's coordinates get multiplied by $m$ and which by $n$ — always remember that the coordinates of the SECOND point $B$ carry the weight $m$ (the part of …