Question

The x-coordinate of a point of the line joining the points P(2,2,1) and Q(5,1,-2) is 4. Find its z-coordinate

Answer 1

Given P(2,2,1) and Q(5, 1,-2)

Let line divide PQ in the ratio k :1 and given x - coordinate of point on the line is 4 so by section formula

${\displaystyle k=\frac{5k+2}{k+1}}$

${\displaystyle 4 =\frac{5k+2}{k+1}}$

k = 2

Now, z-co-ordinate

${\displaystyle z=\frac{-2k+1}{k+1}=\frac{-2\times 2+1}{2+1}=-\frac{3}{3}=-1}$

z = -1

z-coordinate = -1

Answer 2

Let the point R divide PQ in the ratio λ:1. Then, the coordinates of R will be ${\displaystyle \frac{5\lambda +2}{\lambda +1},\frac{\lambda +2}{\lambda +1},\frac{-2\lambda +1}{\lambda +1}}$

It is given that the x-coordinate of R is 4.

Therefore,

${\displaystyle \frac{5\lambda +2}{\lambda +1}=4}$

⇒ 5λ + 2 = 4λ + 4

⇒ 5λ − 4λ = 4 − 2

⇒ λ = 2

Putting λ = 2 in ${\displaystyle \frac{-2\lambda +1}{\lambda +1}}$ we get

z-coordinate of R = ${\displaystyle \frac{-2\lambda +1}{\lambda +1}=\frac{-2\times 2\times 1}{2+1}=\frac{-3}{3}=-1}$

Hence, the z-coordinate of the point is −1.