Question

Find the equation of a line whose gradient is `-1/2` and which passes through a point M where M divides the line segment joining the points A(6, −2) and B(−4, 3) in the ratio 3 : 2.

Answer 1

Given:

Point M(x, y) divides the line segment joining A(6, −2) and B(−4, 3) in the ratio 3 : 2.

Where, m₁ = 3, m₂ = 2, A(x₁, y₁) = (6, −2) and B(x₂, y₂) = (−4, 3),

Substituting the values:

⇒ `x = (3(-4) + 2(6))/(3 + 2)`

= `(-12 +12)/5`

∴ x = 0

⇒ `y = (3(3) + 2(-2))/(3 + 2)`

= `(9 - 4)/5`

= `5/5`

∴ y = 1

The coordinates of point M are (0, 1).

Here, the required line has a gradient (slope) m = `-1/2`,

Using the point–slope formula:

y − y₁ = m(x − x₁)

`y - 1 = -1/2 (x - 0)`

2(y − 1) = −x

2y − 2 = −x

∴ x + 2y − 2 = 0

Hence, the equation of the line is x + 2y − 2 = 0 or x + 2y = 2.