Question

Given a line segment AB joining the points A(–4, 6) and B(8, –3). Find

1) The ratio in which AB is divided by y-axis.

2) Find the coordinates of the point of intersection.

3) The length of AB.

Answer 1

1) Let the line segment AB be divided by the point C on the y-axis in the ratio k: 1.

Then, by section formula, the coordinates of point C are `(8k - 4)/(k + 1), (-3k + 6)/(k + 1)`

Since C lies on the y-axis, coordinates of C are (0, y). On comparing, we have

`(8k - 4)/(k + 1) = 0 => 8k - 4 = 0 => k = 1/2`

Thus, the required ratio in which AB is divided by the y-axis is 1 : 2.

2) The point of intersection of AB and the y-axis is C.

The coordinates of point C are `((8k - 4)/(k + 1), (-3k + 6)/(k + 1)) = ((8xx1/2-4)/(1/2 + ), (-3 xx 1/2 + 6)/(1/2 + 1))`

` = ((4-4)/(1/2 + 1), (-3/2 + 6)/(1/2 + 1))`

= (0,3)

3) By distance formula, `ABsqrt((8+4)^2 + (-3 - 6)^2) = sqrt((12)^2 + (-9)^2) = sqrt225` = 15 units.