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
(a)`a-2`
(b) `a+2`
(c)`2/(a+2)`
(d) `2/a`

Options

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

Answer

3

Explanation

The equation of the given lines are
`y-1=tantheta(x-1)`
and `y-1=cot theta (x-1)` So , their joint equations is
`[(y-1)-tantheta(x-1)][(y-1)-cottheta(x-1)]=0`
or `(y-1)^(2)-(tantheta+cot theta)(x-1)(y-1)+(x-1)^(2)=0`
or `x^(2)-(tantheta+cottheta)xy+y^(2)+(tantheta+cottheta-2)(x+y-1)=0`
Comparing with given equation , we get
`tantheta+cottheta=a+2`
or `(1)/(sinthetacostheta)=a+2`
or `sin2theta=(2)/(a+2)`