If the lines
`hati-(hati+hatj+hatk) xx (1-p)hati+3hatj-2hatk=0`
and `(vecc-(3hati+hatj-5hatk)) xx (1-p)hati+3hatj-2hatk=0`
are coplanar then the value of p is

Options

A. `4/3`
B. 1
C. `2/3`
D. `1/3`

Answer

D

Explanation

Line `vecr-(veci+vecj+veck) xx (1-p)hati+3hatj-2hatk=0` is passing through the point A(1,1,1) and parallel to the vector `vecn_(1)=(1-p)hati+3hatj-2hatk`.
Line `vecr-(3hati+hatj-5hatk) xx (3-p)hati+4hatj-8hatk=0` is passing through the point B(3,1,-5) and parallel to the vector `vecn_(2) = (3-p)hati+4hatj-8hatk`. Since, lines are coplanar, vectors `vecn_(1), vecn_(2)` and `vec(AB)` are coplanar.
`therefore |{:(2,0,-6),(1-p,3,-2),(3-p,4,-8):}|=0`
`rArr p=1/3`