Let the point be
A(5,-1)
B(-1,-4)
p(0,y)
let ratio be k:1
hence, m1=K m2=1
x1=5, y1=-1
x2=-1,y2=-4
x=0,y=y
using section formula.
x=m1x2+m2x1/m1+m2
0=k × -1 + 1×5/ k+1
0=-K+5/K+1
0(K+1)=-K+5
0=-k+5
K=5
now we need to find y also
so,
y=m1y2+m2y1/m1+m2
y=k×-4+1×-1/K+1
y=5×-4+(-1)/5+1
y=-20-1/6
=-21/6= -7/2

Given:
The two points are (5, -1) and (-1, -4).

To Find:
The ratio in which the y-axis divides the line segment joining the given points.

Solution:

Step 1: Find the coordinates of the point where the line segment intersects the y-axis.
Since the y-axis is perpendicular to the x-axis, the x-coordinate of the point where the line segment intersects the y-axis will be 0. Therefore, we need to find the y-coordinate of this point.

To find the y-coordinate, we can use the equation of the line passing through the two given points. The equation of a line passing through two points (x1, y1) and (x2, y2) is given by:

(y - y1) / (x - x1) = (y2 - y1) / (x2 - x1)

Substituting the given points, we have:

(y - (-1)) / (x - 5) = (-4 - (-1)) / (-1 - 5)

Simplifying the equation:

(y + 1) / (x - 5) = (-4 + 1) / (-1 - 5)
(y + 1) / (x - 5) = (-3) / (-6)
(y + 1) / (x - 5) = 1/2

Cross-multiplying and simplifying, we get:

2(y + 1) = x - 5
2y + 2 = x - 5
2y = x - 7
y = (1/2)x - (7/2)

Since the x-coordinate is 0, substitute x = 0 in the equation:

y = (1/2)(0) - (7/2)
y = -7/2

Therefore, the point where the line segment intersects the y-axis is (0, -7/2).

Step 2: Calculate the ratio.
The ratio in which the y-axis divides the line segment can be calculated by dividing the distance between the y-coordinate of the intersection point and the y-coordinate of the first point by the distance between the y-coordinate of the intersection point and the y-coordinate of the second point.

Let's find the distances:

Distance between (5, -1) and (0, -7/2):
d1 = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(0 - 5)^2 + ((-7/2) - (-1))^2]
= √[25 + (-(3/2))^2]
= √[25 + 9/4]
= √(109/4)
= √109 / 2

Distance between (-1, -4) and (0, -7/2):
d2 = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(0 - (-1))^2 + ((-7/2) - (-4))^2]
= √[(1)^2 + (-(1/2))^2]
= √[1 + 1/4]
= √(5/4)
=