Let $z=x+iy$ $\frac{z-i}{z+i}$ is purely imaginary means its real part is zero. $\frac{x+iy-i}{x+iy+i}=\frac{x + i\left(y -1\right)}{x + i\left(y +1\right)}\times\frac{x - i\left(y +1\right)}{x - i\left(y +1\right)}$ $=\frac{x^{2}-2ix\left(y+1\right)+xi\left(y-1\right)+y^{2}-1}{x^{2}+\left(y+1\right)^{2}}$ $=\frac{x^{2}+y^{2}-1}{x^{2}+\left(y+1\right)^{2}}-\frac{2xi}{x^{2}+\left(y+\right)^{2}}$ for pure imaginary, we have $\frac{x^{2}+y^{2}-1}{x^{2}+\left(y+1\right)^{2}}=0$ $\Rightarrow x^{2}+y^{2}=1 \Rightarrow \left(x + iy\right) \left(x - iy\right) = 1$ $\Rightarrow x+iy=\frac{1}{x-iy}=z$ and $\frac{1}{z}=x-iy$ $z+\frac{1}{z}=\left(x+iy\right)+\left(x-iy\right)=2x$ $\left(z+\frac{1}{z}\right)$ is any non-zero real number
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Top Questions on Complex Numbers and Quadratic Equations
Complex Number: Any number that is formed as a+ib is called a complex number. For example: 9+3i,7+8i are complex numbers. Here i = -1. With this we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1.
Quadratic equation: A polynomial that has two roots or is of the degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b and c are the real numbers.
