If points `P(4,5,x),Q(3,y,4) and R(5,8,0)` are collinear, then the value of x+y is

Options

A. `-4`
B. `3`
C. 5
D. 4

Answer

D

Explanation

Given `P(4,5,x),Q(3,y,4) and R(5,8,0)` are collinear.
`therefore vec(PQ)=lamda vec(QR)`
`impliesvec(PQ)=vec(OQ)-vec(OP)`
`=(3hati+lamdahatj+4hatk)-(4hati+5hatj+xhatk)`
`=-hati(6-5)hatj+(4-x)hatk`
and `vec(QR)=vec(OR)-vec(OQ)=(5hati+8hatj)(3hati+yhatj+4hatk)`
`=2hati+(8-y)hatj-4hatk`
`therefore--hati+(y-5)hatj+(4-x)amda[2hati+(8-y)hatj-4hatk]`
On euatting the component of vector both sides, we get
`(-1)/(2)=lamda,(y-5)/(8-y)=lamda,(4-x)/(-4)=lamda`,
On putting the value of `lamda`, we get
`(y-5)=(8-y)(-(1)/(2))`
and `4-x=-4(-(1)/(2))`
`implies2y-10=y-8`
and `4-x=2`
`impliesy=2`
`and x=2`
`thereforex+y=2+2=4`