33. Point $K$ divides the segment $M N$ in the ratio $|M K|:|K N|=$ $=2: 3$. Find the coordinates of point $K$, if $M(7 ; 4) ; N(-3 ; 9)$.
Solution. Substituting $\lambda=2 / 3$ and the coordinates of points $M$ and $N$ into formula (3), we find

$$
\begin{gathered}
x_{K}=\frac{x_{M}+\lambda x_{N}}{1+\lambda}=\frac{7+\frac{2}{3}(-3)}{1+\frac{2}{3}}=\frac{7-2}{\frac{5}{3}}=3 \\
y_{K}=\frac{y_{M}+\lambda y_{N}}{1+\lambda}=\frac{4+\frac{2}{3} \cdot 9}{1+\frac{2}{3}}=\frac{4+6}{\frac{5}{3}}=6 .
\end{gathered}
$$

Thus, $K(3 ; 6)$.