Let the coordinates of the point be (0,y) and the y axis divide the line segment in the ratio k:1

(0,y)={(-k+5)/k+1 , (-4k-6)/k+1}

=>0=-k+5/k+1

=>0=-k+5

=>k=5

y=-4k-6/k+1

=>y=-4(5)-6/5+1

=>y=-20-6/6

=>y=-26/6

thus the coordinates of the point is (0,-26/6)

That's all🙂

**Finding the Ratio in which the Y-axis Divides the Line Segment**

To find the ratio in which the y-axis divides the line segment joining the points (5, -6) and (-1, -4), we need to determine the coordinates of the point of intersection and then calculate the ratio of the distances from each endpoint to the intersection point.

**Step 1: Determine the Equation of the Line Segment**

The equation of the line segment joining the points (5, -6) and (-1, -4) can be found using the slope-intercept form of a linear equation: y = mx + c, where m is the slope and c is the y-intercept.

First, calculate the slope (m) of the line segment using the formula:

m = (y2 - y1) / (x2 - x1)

Substituting the coordinates of the two points:

m = (-4 - (-6)) / (-1 - 5)
= 2 / (-6)
= -1/3

Now, we can determine the y-intercept (c) by substituting the coordinates of one of the points into the equation:

-6 = (-1/3)(5) + c
c = -6 + 5/3
c = -18/3 + 5/3
c = -13/3

So, the equation of the line segment is y = (-1/3)x - 13/3.

**Step 2: Find the Point of Intersection**

To find the point of intersection, we need to determine the value of x when y = 0 (since the y-axis intersects the x-axis at y = 0).

Setting y = 0 in the equation of the line segment:

0 = (-1/3)x - 13/3

Solving for x:

(-1/3)x = 13/3
x = (13/3)(-3)
x = -13

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

**Step 3: Calculate the Ratio of the Distances**

We can now calculate the ratio in which the y-axis divides the line segment by finding the distances from each endpoint to the intersection point (-13, 0).

Distance from (5, -6) to (-13, 0):

d1 = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(-13 - 5)^2 + (0 - (-6))^2]
= √[(-18)^2 + 6^2]
= √[324 + 36]
= √360
= 6√10

Distance from (-1, -4) to (-13, 0):

d2 = √[(-13 - (-1))^2 + (0 - (-4))^2]
= √[(-12)^2 + 4^2]
= √[144 + 16]
= √160
= 4√10

Therefore, the ratio in which the y-axis divides the line segment is:

d1 : d2 = 6√10 : 4√10
= 3√10 : 2√10

Simplifying the ratio, we get:

d1 : d2 =