The problem states:

• The coordinates of the endpoints of $UV$ are $U(1, -6)$ and $V(11, 9)$.
• Point $W$ is on $UV$ and divides it such that the ratio $UW : VW = 3:2$.

We need to find the coordinates of $W$.

Solution Approach:

We can use the section formula. If a point divides a line segment in a ratio $m:n$, the coordinates $(x, y)$ of the dividing point are given by:

$x = \frac{m \cdot x_2 + n \cdot x_1}{m+n}, \quad y = \frac{m \cdot y_2 + n \cdot y_1}{m+n}$

Where:

• $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the endpoints of the line.
• $m:n$ is the given ratio.

In this case:

• $(x_1, y_1) = (1, -6)$ and $(x_2, y_2) = (11, 9)$,
• The ratio $m:n = 3:2$.

Let's now calculate the coordinates of $W$. I'll do the math.