Question

draw in the complex plane the set {z ∈ c: im(1/z)=1/4}

Explanation:

Answer:

Describe the set as a circle in the complex - plane. Let \(z=x + iy\), then \(\frac{1}{z}=\frac{1}{x + iy}=\frac{x-iy}{x^{2}+y^{2}}\). The imaginary part of \(\frac{1}{z}\) is \(-\frac{y}{x^{2}+y^{2}}\). Setting \(-\frac{y}{x^{2}+y^{2}}=\frac{1}{4}\) gives \(x^{2}+y^{2}+4y = 0\), which can be rewritten as \(x^{2}+(y + 2)^{2}=4\). So it is a circle centered at \((0,-2)\) with radius \(2\). Sketch a circle centered at the point \((0,-2)\) in the complex - plane with radius \(2\), label the real and imaginary axes.