If the tangent to the curve `x y+a x+b y=0` at `(1,1)` is inclined at an angle `tan^(-1)2` with x-axis, then find `a and b ?`

Options

A. 0
B. `(1)/(2)`
C. `-(1)/(2)`
D. none of these

Answer

B

Explanation

We have,
` xy+ax+by=0 " "...(i)`
` rArr x(dy)/(dx)+y+a+b(dy)/(dx)=0 `
` rArr (dy)/(dx)=-((y+a)/(x+b)) `
` rArr ((dy)/(dx))_((1","1))=-((a+1)/(b+1)) `
` rArr 2=-((a+1)/(b+1)) " "`[Given : Slope of the tangent at (1, 1) is ` tan theta =2`]
` rArr a+2b=-3 " "...(ii) `
Also, (1, 1) lies on (i).
` therefore a+b=-1 " "...(iii) `
Solving (ii) and (iii), we get
` a=1, b=-2 rArr (a+b)/(ab)=(1)/(2) `