Find the ratio in which the point (−3, k) divides the join of A(−5, −4) and B(−2, 3).

Solution:

Let the point P(−3, k) divide the line AB in the ratio s : 1.
Then, by the section formula:

$x=\frac{m x_{2}+n x_{1}}{m+n}, y=\frac{m y_{2}+n y_{1}}{m+n}$

The coordinates of $P$ are $(-3, k)$.

$-3=\frac{-2 s-5}{s+1}, k=\frac{3 s-4}{s+1}$

$\Rightarrow-3 s-3=-2 s-5, k(s+1)=3 s-4$

$\Rightarrow-3 s+2 s=-5+3, k(s+1)=3 s-4$

$\Rightarrow-s=-2, k(s+1)=3 s-4$

$\Rightarrow s=2, k(s+1)=3 s-4$

Therefore, the point $P$ divides the line $A B$ in the ratio $2: 1$.

Now, putting the value of $s$ in the equation $k(s+1)=3 s-4$, we get:

$k(2+1)=3(2)-4$

$\Rightarrow 3 k=6-4$

$\Rightarrow 3 k=2 \Rightarrow k=\frac{2}{3}$

Therefore, the value of $k=\frac{2}{3}$

That is, the coordinates of $P$ are $\left(-3, \frac{2}{3}\right)$.