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

#### Solution 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

` k = (5k + 2)/(k+1)`

`4 = (5k + 2)/(k+1)`

k = 2

Now, z-co-ordinate

`z = (-2k+1)/(k+1) = (-2xx2+1)/(2+1) = -3/3 = -1`

z = -1

z-coordinate = -1

#### Solution 2

Let the point R divide PQ in the ratio λ:1. Then, the coordinates of R will be `(5lambda+2)/(lambda+1),(lambda+2)/(lambda+1), (-2lambda+1)/(lambda + 1)`

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

Therefore,

`(5lambda+2)/(lambda+1) = 4`

⇒ 5λ + 2 = 4λ + 4

⇒ 5λ − 4λ = 4 − 2

⇒ λ = 2

Putting λ = 2 in `(-2lambda + 1)/(lambda+1)` we get

*z*-coordinate of R = `(-2lambda + 1)/(lambda + 1) = (-2xx2xx1)/(2+1) = (-3)/3 = -1`

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