# Chapter 8 - Review Exercises - Page 410: 41

The graphs of all the complex numbers will be lying on the line $y = - x$ in both the 4th quadrant with trigonometric form $x\sqrt{2}(cos315^\circ + isin315^\circ)$ and in the 2nd quadrant with trigonometric form $x\sqrt{2}(cos135^\circ + isin135^\circ)$ respectively.

#### Work Step by Step

As $z = x + yi$, when the imaginary part of $z$ is the negative of the real part of $z$, $y = - x$ then. The complex number, $z = x + yi$, will become $z = x - xi$. When $x \gt 0$, $z$ is at $315^\circ$ with absolute value $\sqrt{x^2 + (-x)^2} = x\sqrt{2}$, all the complex numbers will be lying on the line $y = - x$ in the 4th quadrant with trigonometric form $x\sqrt{2}(cos315^\circ + isin315^\circ)$. However, when $x \lt 0$, $z$ is at $135^\circ$ with absolute value $\sqrt{(-x)^2 + x^2} = x\sqrt{2}$ same, all the complex numbers will still be lying on the line $y = - x$ but in the 2nd quadrant with trigonometric form $x\sqrt{2}(cos135^\circ + isin135^\circ)$.

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