In what ratio does the point T (3, 0) divide the segment joining the points S (4, –2) and U (1, 4)?

Let point T divide line segment SU in the ratio k :1.

If the co-ordinates of point T be (x, y) and that of points $S$ an $U$ be $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ respectively, then

$x=\frac{k x_{2}+x_{1}}{k+1} ; y=\frac{k y_{2}+y_{1}}{k+1}$

$\therefore \quad 3=\frac{k \times 1+1 \times 4}{k+1}$

$\Rightarrow 3 k+3=k+4$

$\Rightarrow 3 k-k=4-3 \Rightarrow 2 k=1$

$\Rightarrow k=\frac{1}{2}$