Given points $A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right)$ and a non-negative number $\lambda$. Find the coordinates of point $M$ on the ray $A B$, for which $A M: A B=\lambda$.
If the point $M\left(x_{0}, y_{0}\right)$ lies on the line $A B$ and $A M: A B=\lambda$, then by the theorem of proportional segments, the same ratio holds for the projection of point $M$ on the $O X$ axis, i.e., $\frac{x_{0}-x_{1}}{x_{2}-x_{1}}=\lambda$. From this, we find that $x_{0}=(1-\lambda) x_{1}+\lambda x_{2}$.

Similarly, $y_{0}=(1-\lambda) y_{1}+\lambda y_{2}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_198acf192511df7a04ddg-11.jpg?height=478&width=715&top_left_y=1333&top_left_x=680)

## Answer

$\left((1-\lambda) x_{1}+\lambda x_{2},(1-\lambda) y_{1}+\lambda y_{2}\right)$.