If the angle between the line `x=(y-1)/(2)=(z-3)(lambda)` and the plane `x+2y+3z=4 is cos^(-1)(sqrt((5)/(14)))`, then `lambda` equals

Options

A. `2/3`
B. `3/2`
C. `2/5`
D. `5/3`

Answer

A

Explanation

Let `theta` be the angle between the line
`(x-0)/1=(y-1)/2=(z-3)/(lamda)` and the plane `x+2y+3z-4=0`. Then,
`cos((pi)/2-theta)=(1xx1+2xx2+3xxlamda)/(sqrt(1+4+9)sqrt(1+4+lamda^(2))) [ :' cos ((pi)/2 theta)=(vecb.vecn)/(|vecb||vecn|) ]`
`impliessin theta=(5+3lamda)/(sqrt(14)sqrt(5+lamda^(2)))`
`impliessin^(2)theta=((5+3lamda)^(2))/(14(5+lamda^(2)))`
`implies1-5/14=((5+3lamda)^(2))/(14(5+lamda^(2))) [ cos theta=sqrt(5/14) "(given)"]`
`implies45+9lamda^(2)=25+30lamda+9lamda^(2)implieslamda=2/3`