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Area Method to Find Condition for Co-Linearity

lf the comple...

Question

lf the complex numbers $z_{1}, z_{2}$ and $z_{3}$ respectively the points A, B,C on the Argand plane and
$∣ ∣ ∣ ∣ \begin{matrix} z_{1} & z_{2} & 1 z_{2} & z_{3} & 1 z_{3} & z_{1} & 1 \end{matrix} ∣ ∣ ∣ ∣$ =0, then $Δ A B C$ is

scalene

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isosceles

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equilateral

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right-angled isosceles

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Solution

The correct option is C equilateral
We have,
$∣ ∣ ∣ ∣ \begin{matrix} z_{1} & z_{2} & 1 z_{2} & z_{3} & 1 z_{3} & z_{1} & 1 \end{matrix} ∣ ∣ ∣ ∣ = z_{1} (z_{3} - z_{1}) - z_{2} (z_{2} - z_{3}) + 1 (z_{2} z_{1} - z_{3}^{2}) = 0$
$\Rightarrow z_{1} z_{2} + z_{2} z_{3} + z_{1} z_{3} - z_{1}^{2} - z_{2}^{2} - z_{3}^{2} = 0$
which is the $c o n d^{n}$ for veritices of equilateral $Δ$

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