USING THE EQUATION OF LINE TO PROVE THREE POINTS ARE COLLINEAR

By using the concept of equation of line, prove that the three points are collinear.

Example 1 :

(4, 2), (7, 5), (9, 7)

Solution :

Formula for slope of a line joining two points :

m = (y₂ - y₁)/(x₂ - x₁)

Substitute (x₁, y₁) = (4, 2) (x₂, y₂) = (7, 5).

m = (5 - 2)/(7 - 4)

= 3/3

= 1

Equation of line in slope-intercept form :

y = mx + b

Substitute m = 1.

y = x + b ----(1)

The line is passing through the point (4, 2).

So, substitute x = 4 and y = 2.

2 = 4 + b

-2 = b

Substitute b = -2 in (1).

y = x - 2

Now, substitute the third point (9, 7) into the equation of line and check whether the point satisfies the equation.

9 = 7 - 2 ?

7 = 7 ✓

The above result is true. So the point (9, 7) satisfies the equation of line.

Hence, the three points (4, 2), (7, 5) and (9, 7) are collinear.

Example 2 :

(1, 4), (3, -2), (-3, 16)

Solution :

Formula for slope of a line joining two points :

m = (y₂ - y₁)/(x₂ - x₁)

Substitute (x₁, y₁) = (1, 4) (x₂, y₂) = (3, -2).

m = (-2 - 4)/(3 - 1)

= -6/2

= -3

Equation of line in slope-intercept form :

y = mx + b

Substitute m = -3.

y = -3x + b ----(1)

The line is passing through the point (1, 4).

So, substitute x = 1 and y = 4.

4 = -3(1) + b

4 = -3 + b

Substitute b = 7 in (1).

y = -3x + 7

Now, substitute the third point (-3, 16) into the equation of line and check whether the point satisfies the equation.

16 = -3(-3) + 7 ?

16 = 9 + 7 ?

16 = 16 ✓

The above result is true. So the point (-3, 16) satisfies the equation of line.

Hence, the three points (1, 4), (3, -2) and (-3, 16) are collinear.

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