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 the y-axis.
2. find the coordinates of the point of intersection.
3. the length of AB.

Answer 1

i. Let the required ratio be m₁ : m₂

Consider A(−4, 6) = (x₁, y₁); B(8, −3) = (x₂ , y₂) and let

P(x, y) be the point of intersection of the line segment and the y-axis

By section formula, we have,

`x = (m_1x_2 + m_2x_1)/(m_1 + m_2), y = (m_1y_2 + m_2y_1)/(m_1 + m_2)`

`=> x = (8m_1 - 4m_2)/(m_1 + m_2), y = (-3m_1 + 6m_2)/(m_1 + m_2)`

The equation of the y-axis is x = 0

`=> x = (8m_1 - 4m_2)/(m_1 + m_2) = 0`

`=> 8m_1 - 4m_2 = 0`

`=> 8m_1 = 4m_2`

`=> m_1/m_2 = 4/8`

`=> m_1/m_2 = 1/2`

ii. From the previous subpart, we have,

`m_1/m_2 = 1/2`

`=>` m₁ = k and m₂ = 2k, where k

Is any constant.

Also, we have,

`=> x = (8m_1 - 4m_2)/(m_1 + m_2), y = (-3m_1 + 6m_2)/(m_1 + m_2)`

`=> x = (8 xx k - 4 xx 2k)/(k + 2k), y = (-3 xx k + 6xx 2k)/(k + 2k)`

`=> x = (8k - 8k)/(3k), y = (-3k + 12k)/(3k)`

`=>x = 0/(3k), y = (9k)/(3k)`

`=>` x = 0, y = 3

Thus, the point of intersection is p (0, 3)

iii. The length of AB = distance between two points A and B.

The distance between two given points

A(x₁, y₁) and B(x₂, y₂) is given by,

Distance AB = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`

= `sqrt((8 + 4)^2 + (-3 - 6)^2)`

= `sqrt((12)^2 + (9)^2)`

= `sqrt(144 + 81)`

= `sqrt(225)`

= 15 units