# Recent questions and answers in Complex Numbers

Questions from: Complex Numbers

### $P$ represents the variable complex number $z$.Find the locus of $P$,if $Im\bigg[\large\frac{2z+1}{iz+1}\bigg]$$=-2 ### If arg \left ( z-1 \right ) = \large\frac{\pi}{6} and arg \left ( z+1 \right ) = 2\large\frac{\pi}{3} then prove that \left | z \right |=1 ### Express the following complex numbers in polar form. 1 - \mathit{i} ### Express the following complex numbers in polar form. -1 - \mathit{i} ### Express the following complex number in polar form. -1 + \mathit{i}\sqrt{3} ### Express the following complex number in polar form. 2 + 2\sqrt{3}\mathit{i} ### Prove that the points representing the complex numbers \left ( 7+5\mathit{i} \right ), \left ( 5+2\mathit{i} \right ), \left ( 4+7\mathit{i} \right ) and \left ( 2+4\mathit{i} \right ) form a parallelogram. (Plot the points and use midpoint formula). ### Prove that the triangle formed by the points representing the complex numbers (10+8i),(-2+4i) and (-11+31i) on the Argand plane is right angled. ### If z^2=(0,1) find z. ### Find the square root of (-8-6i) ### If \left ( 1+\mathit{i} \right )\left ( 1+2\mathit{i} \right )\left ( 1+3\mathit{i} \right )... \left ( 1+\mathit{ni} \right )=\mathit{x+iy}, show that 2.5.10 ... \left ( 1+n^{2} \right ) = x^{2}+y^{2} ### For what values of \mathit{x} and \mathit{y}, the numbers -3+\mathit{ix^{2}y} and \mathit{x^{2}}+y+4\mathit{i} are complex conjugate of each other? ### Find the real values of \mathit{x} and \mathit{y} for which the following equation: \sqrt{x^{2}+3x+8} + \left ( x+4 \right )\mathit{i} = \mathit{y}\left (2 + \mathit{i} \right ) ### Find the real values of \mathit{x} and \mathit{y} for which the following equation: \large\frac{\left ( 1+\mathit{i} \right )\mathit{x}-2\mathit{i}}{3+\mathit{i}} + \frac{\left ( 2-3\mathit{i} \right )\mathit{y}+\mathit{i}}{3-\mathit{i}} =$$\mathit{i}$

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