Given:
The coordinates of point A are (5, -6).
The coordinates of point B are (-1, -4).

To Find:
1. The ratio in which the y-axis divides the line segment joining points A and B.
2. The coordinates of the point of division.

Solution:

Step 1: Find the Distance between A and B
To find the ratio in which the y-axis divides the line segment AB, we first need to find the distance between points A and B using the distance formula.

The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)

Using the coordinates of points A and B:
d = √((-1 - 5)^2 + (-4 - (-6))^2)
= √((-6)^2 + (2)^2)
= √(36 + 4)
= √40
= 2√10

Step 2: Find the Ratio
The ratio in which the y-axis divides the line segment AB is given by:
Ratio = Distance from A to the point of division / Distance from the point of division to B

Let the coordinates of the point of division be (0, y). Since the point lies on the y-axis, the x-coordinate is 0.

Using the distance formula, we can calculate the distance from A to the point of division:
Distance from A to (0, y) = √((0 - 5)^2 + (y - (-6))^2)
= √(25 + (y + 6)^2)
= √(y^2 + 12y + 61)

Similarly, the distance from the point of division to B is:
Distance from (0, y) to B = √((-1 - 0)^2 + (-4 - y)^2)
= √(1 + (y + 4)^2)
= √(y^2 + 8y + 17)

Therefore, the ratio is:
Ratio = √(y^2 + 12y + 61) / √(y^2 + 8y + 17)

Step 3: Solve for the Point of Division
To find the coordinates of the point of division, we need to determine the value of y that satisfies the given ratio.

By comparing the ratio with the given ratio, we have the equation:
√(y^2 + 12y + 61) / √(y^2 + 8y + 17) = given ratio

Squaring both sides of the equation, we get:
(y^2 + 12y + 61) / (y^2 + 8y + 17) = given ratio^2

Cross-multiplying the equation, we have:
(y^2 + 12y + 61) = (y^2 + 8y + 17) * given ratio^2

Expanding and rearranging the equation, we get a quadratic equation in terms of y:
(y^2 + 12y + 61) - (y^2 + 8y + 17) * given ratio^2 = 0

Simplifying

(0,-13/3)