The problem asks to find the coordinates of points $C$, $D$, and $B$, given the coordinates of point $A$, and some ratio conditions for line segments. Let's break it down:

Given:

• Coordinates of $A(-2, 6)$.
• Line segment $AB$ is intercepted by the $y$-axis at $C$ and the $x$-axis at $D$.
• The ratio $AC : AD = 1 : 4$, meaning $\frac{AC}{AD} = \frac{1}{4}$.
• $D$ is the midpoint of $CB$.

We can solve this step by step using the section formula and midpoint formula:

Step 1: Find coordinates of point $D$

Since $D$ lies on the $x$-axis, its coordinates will be $D(x_1, 0)$. We use the section formula in the ratio $1 : 4$ to find the coordinates of $D$.

The section formula for dividing a line segment joining $(x_1, y_1)$ and $(x_2, y_2)$ in the ratio $m:n$ is:

$(x, y) = \left( \frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n} \right)$

Here, point $C(0, y_C)$ lies on the $y$-axis (as it is the y-intercept). Applying the section formula to divide the segment $AC$:

[ x_D = \frac{1 \cdot x_D + 4 \cdot (-2)}{1 + 4} = \frac{-8}{5}