Question

Condition For co linear points ........ Co ordinate geometry ......​

Answer 1

Condition of collinearity of three points :

Here, we are going to see the conditionn of collinearity of three points.

If three points are collinear, then they must lie on the same line.

We can prove the collinearity of three points using one of the below three concepts.

(i) Concept of slope

(ii) Concept of distance between the two points

(iii) Concept of area of triangle

(iv) Concept of equation of line

Concept of slope

Let A, B and C be the three points.

If we want A, B and C be collinear, the following conditions have to be met.

(i) Slope of AB = Slope of BC

(ii) There must be a common point between AB and BC.

(In AB and BC, the common point is B)

If the above two conditions are met, then the three points A, B and C are collinear.

Concept of distance between the two points

Let A, B and C be the three points.

We have to find the three lengths AB, BC and AC among the given three points A, B and C.

The three points A, B and C are collinear, if the sum of the lengths of any two line segments among AB, BC and AC is equal to the length of the remaining line segment.

That is,

AB + BC = AC

(or)

AB + AC = BC

(or)

AC + BC = AB

Concept of area of triangle

Let A(x₁, y₁), B(x₂, y₂) and C (x₃, y₃) be the three points.

If the three points, A, B and C are collinear, they will lie on the same and they cannot form a triangle.

Hence, the area of triangle ABC = 0

1/2 x { (x₁y₂ + x₂y₃ + x₃y₁) - (x₂y₁ + x₃y₂ + x₁y₃) } = 0

(or)

x₁y₂ + x₂y₃ + x₃y₁ = x₂y₁ + x₃y₂ + x₁y₃

Concept of equation of line

Let A(x₁, y₁), B(x₂, y₂) and C (x₃, y₃) be the three points.

Let us find the equation through any two of the given three points. If the third point satisfies the equation, then the three points A, B and C will be collinear.

$\huge \mathrm \pink{karnataka}$

Answer 2

Answer:

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