If `z_(1) and z_(2)` are z-coordinates of the points of trisection of the segment joining the points A(2,1,4), B(-1,3,6_, then `z_(1)+z_(2)+`

Options

A. 1
B. 4
C. 5
D. 10

Answer

D

Explanation

Let `z_(1)` and `z_(2)` be the z-coordinates of points P and Q, respectively which trisects the line segment AB.

Now, coordinates of
`P = ((1 xx -1 + 2 xx2)/(1 + 2), (1xx3 + 2 xx 1)/(1 + 2) , (1 xx 6 + 2 xx 4)/(1 + 2))`
[by internal section formula]
`:.` z-coordinate of `P = (1 xx 6 + 2 xx 4)/( 1 + 2) `
`rArr z_1 = (6 + 8 )/(3) = 14/3 "...."(i)`
[z-coordinate of P = `z_(1)` (given)]
Again, coordinates of
`Q = ((2 xx - 1 + 1 xx 2)/(2 + 1), (2 xx 3 + 1 xx1)/(2 + 1) , (2 xx 6 + 1 xx 4)/(2 + 1))`
`:. z`-coordinates of `Q = (2 xx 6 + 1 xx 4)/(2 + 1)`
`rArr z_(2) = (12 + 4)/(3) = 16/3"..."(ii)`
[ `:'` z-coordinates of `Q = z_(2)` (given)]
On adding Eqs. (i) and(ii), we get
`z_(1) + z_(2) = (14)/(3) + (16)/(3) = 30/3 = 10`