Q. 44.0( 51 Votes )

Using section formula, show that the points A (2, –3, 4), B (–1, 2, 1) and are collinear.

Answer :

Let the point P divides AB in the ratio k : 1.

Then,


Comparing this information with the details given in the question, we have


x1 = 2, y1 = -3, z1 = 4; x2 = -1, y2 = 2, z2 = 1 and m = k, n = 1


By section formula,


We know that the coordinates of the point R which divides the line segment joining two points P (x1, y1, z1) and Q (x2, y2, z2) internally in the ratio m : n is given by:



So, we have,


The coordinates of P =


Now, we check if for some value of k, the point coincides with the point C.


Put


-k + 2 = 0 k = 2


When k =2, then


And


Therefore, C is a point which divides AB in the ratio 2 : 1 and is same as P.


Hence, A, B, C are collinear.


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