If the angle `theta` between the line `(x+1)/1=(y-1)/2=(z-2)/2` and the plane `2x-y+sqrt(pz)+4=0` is such that `sintheta=1/3`, then the values of p is (A) 0 (B) `1/3` (C) `2/3` (D) none of these`

Options

A. `(-3)/(5)`
B. `(5)/(3)`
C. `(-4)/(3)`
D. `(3)/(4)`

Answer

b

Explanation

The line is `(x+1)/(1)=(y-1)/(2)=(z-2)/(2)` and the plane is `2x-y+sqrtlamdaz+4=0`.
If `theta` is the angle between the line and the plane, then `90^(@)-theta` is the angle between the line and normal to the plane. Thus,
`cos(90^(@)-theta)=((1)(2)+(2)(-1)+(2)(sqrtlamda))/(sqrt(1+4+4)sqrt(4+1+lamda))`
or `sintheta=(2-2+2sqrtlamda)/(3sqrt5+lamda)or(1)/(3)=(2sqrtlamda)/(3sqrt5+lamda)`
or `sqrt(5+lamda)=2sqrtlamda`
or `5+lamda=4lamda`
or `3lamda=5`
or `lamda=(5)/(3)`