The direction ratios o f a normal to the plane through (1,0,0) and (0,1,0), which makes an angle of `(pi)/(4)` with the plane x+y=3, are

Options

A. `1,sqrt(2),1`
B. `1,1,sqrt(2)`
C. `1,1,2`
D. `sqrt(2),1,1`

Answer

B

Explanation

Let the direction ratios of the normal to the plane be proportional to a,b,c. Then, the equation of the plane is
`a(x-1)+b(y-0)+c(z-0)=0`……………i
It passes through (0,1,0)
`:.a(-1)+b(1)+c(0)=0impliesa=b`…………….ii
It is given that the plane i makes an angle `pi//4` with the plane `x+y=3`.
`:."cos"(pi)/4=(axx1+bxx1+cxx0)/(sqrt(a^(2)+b^(2)+c^(2))sqrt(1+1))`
`implies1/(sqrt(2))=(a+b)/(sqrt(a^(2)+b^(2)+c^(2))sqrt(2))`
`impliessqrt(a^(2)+b^(2)+c^(2))=a+b`
`impliesa^(2)+b^(2)+c^(2)=a^(2)+b^(2)+2abimpliesc^(2)=2ab`............iii
From ii and iii, we have
`a:b:c=a:a:sqrt(2)a=1:1:sqrt(2)`