Question

Complex numbers question
Please give proper explaination

Answer 1

$let \: \: z = x + iy. \: \: then \\ \frac{2z + 1}{iz + 1} = \frac{2(x + iy) + 1}{i(x + iy)+ 1} \\ = \frac{2x + 1 + i2y}{1 - y + ix} \\ = \frac{(2x + 1 + i2y)(1 - y - ix)}{(1 - y + ix)(1 - y - ix)} \\ = \frac{(2x + 1)(1 - y) + 2xy + i(2y(1 - y) - x(2x + 1))}{ {(1 - y)}^{2} + {x}^{2} } \\ \\ given \: \: im( \frac{2z + 1}{iz + 1} ) = 3\\ so \: \: \frac{2y(1 - y) - x(2x + 1)}{{(1 - y)}^{2} + {x}^{2} } = 3 \\ 2y - 2 {y}^{2} - 2 {x}^{2} - x = 3 - 6y + 3 {y}^{2} + 3 {x}^{2} \\ 5 {x}^{2} + 5 {y}^{2} + x - 8y + 3 = 0 \\ \\ this \: \: represents \: \: circle \\ option. \: A \:$