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If the imaginary part of $\frac{2 z + 1}{i z + 1}$ is $- 4,$ then the locus of the point representing z in the complex plane is

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detailed solution

Correct option is C

Let z=x+iy then2z+1iz+1=2(x+iy)+1i(x+iy)+1=(2x+1)+2iy(1−y)+ix=[(2x+1)+2iy][(1−y)−ix](1−y)2+x2As Im⁡2z+1iz+1=−4, we get2y(1−y)−x(2x+1)x2+(1−y)2=−4⇒ 2x2+2y2+x−2y=4x2+4y2−2y+1⇒ 2x2+2y2−x−6y+4=0which represents a circle.

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If the imaginary part of 2z+1iz+1 is -4, then the locus of the point representing z in the complex plane is

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If the imaginary part of $\frac{2 z + 1}{i z + 1}$ is $- 4,$ then the locus of the point representing z in the complex plane is