Q49 / 120 · VITEEE 2012 · Mathematics

If the complex numbers $\mathrm{Z}_{1}, \mathrm{z}_{2}$ and $\mathrm{z}_{3}$ are in $\mathrm{AP}$, then they lie on a

  • A
    a circle
  • B
    a parabola
  • Correct answer
    C
    line
  • D
    ellipse

Solution

Let $\mathrm{z}_{1}, \mathrm{z}_{2}$ and $\mathrm{z}_{3}$ be affixes of points $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$, respectively. Since, $\mathrm{z}_{1}, \mathrm{z}_{2}$ and $\mathrm{z}_{3}$ are in AP, therefore $$ \begin{array}{l} 2 z_{2}=z_{1}+z_{3} \\ \Rightarrow z_{2}=\frac{z_{1}+z_{3}}{2} \end{array} $$ So, $\mathrm{B}$ is the mid-point of the line $\mathrm{AC}$. $\Rightarrow \mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are collinear. $\therefore z_{1}, z_{2}$ and $z_{3}$ lie on a line.

Hence, the correct answer is option: 'C'.

More Complex Number Questions