In which ratio the y-axis divides the line segment joining the points (5, – 6) and (–1, – 4)?

To find the ratio in which the y-axis divides the line segment, we need to find the point where the line segment intersects the y-axis. The x-coordinate of this point will be 0 because any point on the y-axis has an x-coordinate of 0.

The equation of the line passing through the points (5, -6) and (-1, -4) can be found using the formula for the slope of a line:

m = (y2 - y1) / (x2 - x1) = (-4 - (-6)) / (-1 - 5) = 2 / -6 = -1/3

So, the equation of the line is y - y1 = m(x - x1), or y + 6 = -1/3(x - 5).

To find the y-coordinate of the point where this line intersects the y-axis, we set x = 0 in this equation:

y + 6 = -1/3(0 - 5) = 5/3

So, y = 5/3 - 6 = -13/3

Therefore, the line segment intersects the y-axis at the point (0, -13/3).

Now, we can find the distances from this point to the two given points to find the ratio in which the y-axis divides the line segment.

The distance from (0, -13/3) to (5, -6) is sqrt((5 - 0)^2 + (-6 - (-13/3))^2) = sqrt(25 + 1/9) = sqrt(226/9) = 15/3 = 5

The distance from (0, -13/3) to (-1, -4) is sqrt((-1 - 0)^2 + (-4 - (-13/3))^2) = sqrt(1 + 25/9) = sqrt(34/9) = sqrt(34)/3

Therefore, the y-axis divides the line segment in the ratio 5 : sqrt(34)/3.