If the equation of the pair of straight lines passing through the point `(1,1)` , one making an angle `theta` with the positive direction of the x-axis and the other making the same angle with the positive direction of the y-axis, is `x^2-(a+2)x y+y^2+a(x+y-1)=0,a!=2,` then the value of `sin2theta` is

Options

A. `a-2`
B. `a+2`
C. `2//(a+2)`
D. `2//a`

Answer

C

Explanation

Equations of lines passing through `(1,1)` are `y-1=tantheta(x-1)`
and `(y-1cot theta(x-1)`
So, their joint equations is `[(y-1)-tantheta(x-1][(y-1)-cot theta(x-1)]=0`
`rArr x^2-(tantheta+cot theta)xy+y^2+(tantheta+cot theta-2)(x+y-1)-2y-2x+2=0`
Comparing with the given equation , we get `tantheta+cottheta=a+2`
`rArr (sin^2theta+cos^2theta)/(sinthetacostheta)=a+2rArr (1)/(sin2theta)xx2=a+2`
`therefore sin 2 theta=(2)/(a+2)` .