Find the ratio in which the point P (6, k) divides the line segment joining the points M (4, 2) and N (8, -4) . Hence…

Let the point P divides the line segment joining M and N in the ratio $r : 1$.
Then, Coordinates of P = $(\frac{r \times 8 + 1 \times 4}{r+1}, \frac{r \times (-4) + 1 \times 2}{r+1}) = (6, k)$ (1 Mark)
$\Rightarrow \frac{8r + 4}{r+1} = 6$ ($\frac{1}{2}$ Mark)
$\Rightarrow r = 1$ ($\frac{1}{2}$ Mark)
Thus, the point P divides the line segment joining M and N in the ratio $1:1$. ($\frac{1}{2}$ Mark)
i.e. Coordinates of P = $(\frac{4+8}{2}, \frac{2-4}{2}) = (6, k)$ ($\frac{1}{2}$ Mark)
The value of $k$ is $-1$. ($\frac{1}{2}$ Mark)