Complex Numbers
Line in complex plane; slope
Grade Class 12
Question:
Given that $\alpha\bar{z}+\bar{\alpha}z+\gamma=0$ is equation of a line where $\gamma$ is purely imaginary. If $\alpha^2-\beta^2=2$, where $\text{Re}(\alpha)$ and $\text{Im}(\alpha)$ are whole numbers, then slope of the line is
0
$\dfrac{3}{5}$
$-\dfrac{3}{5}$
none of these
Step-by-Step Solution
Key Concept: For whole number Re$(\alpha)$, Im$(\alpha)$: |$\alpha$|$^2=x^2+y^2$. From the constraint, $x^2+y^2=4$ with $x,y\in\mathbb{Z}_+$: only $(x,y)=(2,0)$. So $\alpha=2$ (real), and slope of line $\alpha\bar z+\bar\alpha z=0$ is $0$.
$\alpha=2$, Im$(\alpha)=0$. Slope $=0$.
Correct Answer: 1