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

x_{1} = 2, y_{1} = -3, z_{1} = 4; x_{2} = -1, y_{2} = 2, z_{2} = 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 (x_{1}, y_{1}, z_{1}) and Q (x_{2}, y_{2}, z_{2}) 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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