If the imaginary part of `(2z+1)//(i z+1)` is -2, then find the locus of the point representing in the complex plane.

Explanation

Let x + iy
`rArr (2z +1)/(iz +1)= (2(x +iy)+1)/(i(x+iy)+1)`
`=((2x +1)+i2y)/(1-y +ix)`
`= ((2x+1)+i2y)/((1-y)+ix)((1-y)-ix)/((1-y)-ix)`
`=((2x+1)(1-y)+2xy +i[-x(2x+1)+2y (1-y)])/((1-y)^(2)+x^(2))`
Since imaginary part of `(2z +1)//(iz +1)` is `-2`, we have
`(-x(2x +1)+2y(1-y))/((1-y)^(2) +x^(2))=-2`
or ` -2x^(2) -x+2y _2y^(2) = - 2[1+y^(2) -2y+x^(2)]`
or `x +2y -2=0`, which is a straight line