If the complex numbers $z_{1}, z_{2} a n d z_{3}$ are in AP, then they lie on a.

A circle

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A parabola

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Line

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Ellipse

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Solution

The correct option is C Line
Let $z_{1}, z_{2} a n d z_{3}$ be affixes of points A, B and C respectively. Since, $z_{1}, z_{2} a n d z_{3}$ in AP, therefore
$2 z_{2} = z_{1} + z_{3}$
$\Rightarrow z_{2} = \frac{z_{1} + z_{3}}{2}$
So, B is the mid-point of the line AC.
$\Rightarrow$ A, B and C are collinear.
$∴ z_{1}, z_{2} a n d z_{3}$ lie on a line.

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