What is the condition of collinearity of three points?
We will find the condition of collinearity of three given points by using the concept of slope.
Let P(x\(_{1}\), y\(_{1}\)) , Q (x\(_{2}\), y\(_{2}\)) and R (x\(_{3}\), y\(_{3}\)) are three given points. If the points P, Q and R are collinearity then we must have,
Slop of the line PQ = slop of the line PR
Therefore, \(\frac{y_{1} - y_{2}}{x_{1} - x_{2}}\) = \(\frac{y_{1} - y_{3}}{x_{1} - x_{3}}\)
⇒ (y\(_{1}\) - y\(_{2}\)) (x\(_{1}\) - x\(_{3}\)) = (y\(_{1}\) - y\(_{3}\)) (x\(_{1}\) - x\(_{3}\))
⇒ x\(_{1}\) (y\(_{2}\) - y\(_{3}\)) + x\(_{2}\) (y\(_{3}\) - y\(_{1}\)) + x\(_{3}\) (y\(_{1}\) - y\(_{2}\)) = 0
Which is the required condition of collinearity of the points P, Q and R.
Solved examples using the concept of slope to find the
condition of collinearity of three given points:
1. Using the method of slope, show that the points P(4, 8), Q (5, 12) and R (9, 28) are collinear.
Solution:
The given three points are P(4, 8), Q (5, 12) and R (9, 28).
If the points P, Q and R are collinear then we must have,
x\(_{1}\) (y\(_{2}\) - y\(_{3}\)) + x\(_{2}\) (y\(_{3}\) - y\(_{1}\)) + x\(_{3}\) (y\(_{1}\) - y\(_{2}\)) = 0, where x\(_{1}\) = 4, y\(_{1}\) = 8, x\(_{2}\) = 5, y\(_{2}\) = 12, x\(_{3}\) = 9 and y\(_{3}\) = 28
Now, x\(_{1}\) (y\(_{2}\) - y\(_{3}\)) + x\(_{2}\) (y\(_{3}\) - y\(_{1}\)) + x\(_{3}\) (y\(_{1}\) - y\(_{2}\))
= 4(12 - 28) + 5(28 - 8) + 9(8 - 12)
= 4(-16) + 5(20) + 9(-4)
= -64 + 100 - 36
= 0
Therefore, the given three points P(4, 8), Q (5, 12) and R (9, 28) are collinear.
2. Using the method of slope, show that the points A (1, -1), B (5, 5) and C (-3, -7) are collinear.
Solution:
The given three points are A (1, -1), B (5, 5) and C (-3, -7).
If the points A, B and C are collinear then we must have,
x\(_{1}\) (y\(_{2}\) - y\(_{3}\)) + x\(_{2}\) (y\(_{3}\) - y\(_{1}\)) + x\(_{3}\) (y\(_{1}\) - y\(_{2}\)) = 0, where x\(_{1}\) = 1, y\(_{1}\) = -1, x\(_{2}\) = 5, y\(_{2}\) = 5, x\(_{3}\) = -3 and y\(_{3}\) = -7
Now, x\(_{1}\) (y\(_{2}\) - y\(_{3}\)) + x\(_{2}\) (y\(_{3}\) - y\(_{1}\)) + x\(_{3}\) (y\(_{1}\) - y\(_{2}\))
= 1{5 - (-7)} + 5{(-7) - (-1)} + (-3){(-1) - 5)}
= 1(5 + 7) + 5(-7 + 1) - 3(-1 - 5)
= 1(12) + 5(-6) - 3(-6)
= 12 - 30 + 18
= 0
Therefore, the given three points A (1, -1), B (5, 5) and C (-3, -7) are collinear.
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