Question

Show that the points (1, 4), (3, −2), and (−3, 16) are collinear. Find the equation of the line through them.

Answer 1

The formula for the slope (m) between two points (x₁, y₁) and (x₂, y₂) is:

`m = (y_2 - y_1)/(x_2 - x_1)`

⇒ Slope of line segment AB:

`m_(AB) = (-2 - 4)/(3 - 1)`

= `(-6)/2`

∴ m_AB = −3

⇒ Slope of line segment BC:

`m_(BC) = (16 - (-2))/(-3 - 3)`

= `18/-6`

∴ m_BC = −3

Here, Slope of AB = Slope of BC,

So, it shares a common point B, and collinear points A, B, and C.

Using the point–slope formula:

y − y₁ = m(x − x₁)

y − 4 = −3(x − 1)

y − 4 = −3x + 3

Let’s write the above equation in standard form (Ax + By + C = 0)

3x + y − 4 − 3 = 0

∴ 3x + y − 7 = 0

Hence, the points are collinear as the slopes between them are equal (−3), and the equation of the line that passes through them is 3x + y − 7 = 0.