A point R with x-coordinate 4 lies on the line segment joining the points P (2, –3, 4) and Q (8, 0, 10). Find the coordinates of the point R.

Asked by Pragya Singh | 11 months ago |  168

##### Solution :-

Given

The coordinates of the points P (2, -3, 4) and Q (8, 0, 10).

x1 = 2, y1 = -3, z1 = 4;

x2 = 8, y2 = 0, z2 = 10

Let the coordinates of the required point be (4, y, z).

So now, let the point R (4, y, z) divides the line segment joining the points

P (2, -3, 4) and Q (8, 0, 10) in the ratio k: 1.

By using 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:

$$\dfrac{mx_2+nx_1}{m+n}, \dfrac{my_2+ny_1}{m+n}, \dfrac{mz_2+nz_1}{m+n}$$

So, the coordinates of the point R are given by

$$\dfrac{8k+2}{k+1}, \dfrac{-3}{k+1}, \dfrac{10k+4}{k+1}$$

So, we have

$$\dfrac{8k+2}{k+1}, \dfrac{-3}{k+1}, \dfrac{10k+4}{k+1}=(4,y,z)$$

$$\dfrac{8k+2}{k+1}=4$$

8k + 2 = 4 (k + 1)

8k + 2 = 4k + 4

8k – 4k = 4 – 2

4k = 2

k = $$\dfrac{2}{4}$$

$$\dfrac{1}{2}$$

Now let us substitute the values, we get

So, the coordinates of the point R are,

$$(4, \dfrac{-3}{\dfrac{1}{2}+1},\dfrac{10(\dfrac{1}{2})+4}{\dfrac{1}{2}+1})$$

= (4,-2,6)

Answered by Abhisek | 11 months ago

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