If Im [(z-1)/(2z+1)] = -4, then locus of z is (1) an ellipse (2) a parabola (3) a straight line (4) a circle

Solution:

Given (z-1)/(2z+1) = 4 

(z-1)/(2z+1) = (x+iy-1)/(2(x+iy)+1)

= ((x-1)+iy)/(2x+1)+2iy

Multiply numerator and denominator with (2x+1)-2iy)

= ((x-1)+iy)((2x+1)-2iy)/((2x+1)+2iy)(2x+1-2iy)

= (x-1)(2x+1)+2y2+iy(-2x+2+2x+1)/((2x+1)2+4y2)

Img ((z-1)/(2z+1)) = 3y/(2x+1)2+4y2 = -4

3y = -4(4x2+1+4x+4y2)

16x2+16y2+16x+3y+4 = 0

This is the locus of a circle.

Hence option (2) is the answer.

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