To solve the problem, we will use the section formula, which helps find the coordinates of a point dividing a line segment in a given ratio.

Here, the y-axis divides the line joining $A(5, -6)$ and $B(-1, -4)$. Since the y-axis divides the line segment, the x-coordinate of the dividing point is 0. Let the ratio in which the y-axis divides the segment be $m:n$.

The section formula gives the x-coordinate as: $x = \frac{m \cdot x_2 + n \cdot x_1}{m + n}$

Given that $x = 0$: $0 = \frac{m \cdot (-1) + n \cdot 5}{m + n}$

Simplify: $0 = -m + 5n \quad \text{(numerator must be zero)}$

$m = 5n$

So, the ratio $m:n = 5:1$.

Final Answer:

(b) 5:1

Let me know if you want further clarifications or have more questions! Here are some relative questions for you:

1. What is the section formula and how does it work in 2D geometry?
2. Can we find the coordinates of the dividing point along with the ratio?
3. What happens if the y-axis divides a segment not passing through the origin?
4. How can this problem be extended to 3D space?
5. How does symmetry affect division ratios?

Tip: When the y-axis divides a line segment, the x-coordinate of the dividing point is always 0!