Let `P M` be the perpendicular from the point `P(1,2,3)` to the `x-y` plane. If ` vec (O P)` makes an angle `theta` with the positive direction of the `z-` axis and ` vec (O M)` makes an angle `varphi` with the positive direction of `x-` axis, where `O` is the origin and `theta` and `varphi` are acute angels, then If `f(x+y)=f(x)+f(y)-x y-1`, `AA x , `y in R ` and `f(1)=1,` then the number of solution of `f(n)=n` , `n in N ,` is

Options

A. `costhetacosphi=1/sqrt14`, `tanphi=2`, `tantheta=sqrt5/3`
B. `sinthetasinphi=2/sqrt14`, `tanphi=2`, `tantheta=sqrt5/3`
C. `tanphi=2`, `costhetacosphi=1/sqrt14`, `tantheta=sqrt5/3`
D. `tantheta=sqrt5/3`, `costhetacosphi=1/sqrt14`, `tanphi=2`

Answer

B

Explanation

If `P` be `(x, y, z),` then from the figure,
`x=r sin theta cosphi, y = rsintheta sin phi and z= rcostheta`
`1=rsintheta cosphi, 2= r sintheta sinphi and 3 = rcostheta`

`rArr" "1^(2)+ 2^(2)+3^(2)= r^(2) or r= pm sqrt(14)`
`therefore" "sinthetacosphi = (1)/(sqrt(14)), sin theta sin phi`
`" "=(2)/(sqrt(14)) and cos theta = (3)/(sqrt(14)) `
(neglecting negative sign as `theta and phi` are acute)
`" "(sinthetasinphi)/(sinthetacosphi)=(2)/(1) or tanphi =2`
Also, `tan theta= sqrt(5)//3`.