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
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.
