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=((1xx-1+2xx2)/(1+2),(1xx3+2xx1)/(1+2),(1xx6+2xx4)/(1+2))` [by internal section formula]
`therefore`z-coordinate of `P=(1xx6+2xx4)/(1+2)`
`impliesz_(1)=(6+8)/(3)=(14)/(3)` . . . (i) [z-coordinate of P=`z_(1)` (given)]
Again, coordinates of
`Q=((2xx-1+1xx2)/(2+1),(2xx3+1xx1)/(2+1),(2xx6+1xx4)/(2+1))`
`therefore` z-coordinate of `Q=(2xx6+1xx4)/(2+1)`
`impliesz_(2)=(12+4)/(3)=(16)/(3)` . . . (ii) [`because`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`