Introduction:
In this problem, we are given two points on a Cartesian plane, (5, -6) and (-1, -4), and we need to find the ratio in which the y-axis divides the line segment joining these two points.

Approach:
To find the ratio in which the y-axis divides the line segment, we need to determine the coordinates of the point where the line segment intersects the y-axis. Let's denote this point as (0, y).

Step 1: Find the slope of the line:
Using the formula for the slope of a line, we can find the slope (m) of the line segment joining the two points:
m = (y2 - y1) / (x2 - x1)

Substituting the given coordinates:
m = (-4 - (-6)) / (-1 - 5)
m = (-4 + 6) / (-1 - 5)
m = 2 / (-6)
m = -1/3

Step 2: Use the point-slope form of a line:
Using the point-slope form of a line, we can write the equation of the line passing through the point (5, -6) and having a slope of -1/3:
y - y1 = m(x - x1)

Substituting the values:
y - (-6) = -1/3(x - 5)
y + 6 = -1/3(x - 5)
3(y + 6) = -x + 5
3y + 18 = -x + 5
3y = -x - 13
y = -1/3x - 13/3

Step 3: Find the y-coordinate of the intersection point:
To find the y-coordinate of the point where the line intersects the y-axis, we substitute x = 0 into the equation of the line:
y = -1/3(0) - 13/3
y = -13/3

Therefore, the point of intersection is (0, -13/3).

Step 4: Find the ratio:
To find the ratio in which the y-axis divides the line segment, we need to calculate the distance between the two points and the distance between the intersection point and the second given point.

Distance between (5, -6) and (-1, -4):
d1 = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((-1 - 5)^2 + (-4 - (-6))^2)
= sqrt((-6)^2 + (2)^2)
= sqrt(36 + 4)
= sqrt(40)
= 2sqrt(10)

Distance between (0, -13/3) and (-1, -4):
d2 = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((-1 - 0)^2 + (-4 - (-13/3))^2)
= sqrt((-1)^2 + (-4 + 13/3)^2)
= sqrt(1 + (1/3)^2)
= sqrt(1 + 1/9)
= sqrt(10/9)
=