The line passing through the points `(5, 1, a) and (3, b, 1)` crosses the YZ-plane at the point `(0, (17)/(2), -(13)/(2))`. Then,

Options

A. `a=6,b=4`
B. `a=8,b=2`
C. `a=2,b=8`
D. `a=4,b=6`

Answer

A

Explanation

The equation of the line passing through `(5,1,a)` and `(3,b,1)` is
`(x-5)/(-2)=(y-1)/(b-1)=(z-a)/(1-a)`
The coordinates of any point on this line are
`(-2r+5,(b-1)r+1,(1-a)r+a)`
If it lies on yz-plane. Then,
`-2r+5=0impliesr=5/2`
Thus, the line cuts yz-plane at `(0,(5b-3)/2,(5-3a)/2)`
But the coordinates of this point are given as `(0,17/2,(-13)/2)`
`:.(5b-3)/2=17/2` and `(5-3a)/2=(-13)/2impliesb=4` and `a=6`
ALITER The equation of the line passing through `(5,1,a)` and `(3,b,1)` is
`(x-5)/(-2)=(y-1)/(b-1)=(z-a)/(1-a)`
It cuts yz-plane at `(0,17/2,(-13)/2)`. This means that this point lies on the line.
`:. 5/2=(17/1-1)/(b-1)=((-13)//2-a)/(1-a)`
`impliesb-1=3`an `-13-a=5(1-a)impliesb=4` and `a=6`