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.
`:.vec(PQ)= lambda vec(QR)`
`rArrvec(PQ)=vec(OQ)-vec(OP)`
`=(3hat(i)+yhat(j)+ 4 hat(k))-(4hat(i)+5hat(j)+x hat(k))`
`=-hat(i)+(y-5)hat(j)+(4-x)hat(k)`
and `vec(OR)=vec(OR)-vec(OQ)=(5hat(i)+8hat(j))-(3hat(i)+y hat(j)+ 4 hat(k))`
`=2hat(i)+(8-y)hat(j)- 4hat(k)`
`:.-hat(i)+(y-5)hat(j)+(4-x)hat(k)`
`= lambda[2 hat(i)+(8-y)hat(j)-4hat(k)]`
On equating the component of vector both sides, we get
`(-1)/(2)=lambda, (y-5)/(8-y)=lambda, (4-x)/(-4)=lambda`
On putting the value of `lambda`, we get
`(y-5)=(8-y)(-(1)/(2))`
and `4-x=-4(-(1)/(2))`
`rArr 2y-10=y-8`
and `4-x=2`
`rArr y=2`
and `x=2`
`:. x+y=2+2=4`