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Definition of a Determinant

The necessary...

Question

The necessary and sufficient condition for the points $z_{1}$ , $z_{2}$ , $z_{3}$ to be collinear is that

A

\frac{z_{3} - z_{2}}{z_{2} - z_{1}}

is purely real

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B

\frac{z_{2} + z_{3}}{z_{2} + z_{1}}

is purely real

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C

\frac{z_{3} + z_{1}}{z_{3} + z_{2}}

is purely imaginary

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D

\frac{z_{2} - z_{1}}{z_{3} - z_{2}}

is purely imaginary

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Solution

The correct option is D $\frac{z_{3} - z_{2}}{z_{2} - z_{1}}$ is purely real
Let $z_{1} = r_{1} e^{i θ_{1}}, z_{3} = r_{3} e^{i θ_{3}}$
& $z_{2}$ be origin
$\frac{z_{3} - z_{2}}{z_{2} - z_{1}}$
$= \frac{r_{3} e^{i θ_{3}}}{- r_{1} e^{i θ_{1}}}$
$= \frac{r_{3}}{- r_{1}} e^{i (θ_{3} - θ_{1})}$
For the points to be collinear $(θ_{3} - θ_{1}) = 0$
$\frac{z_{3} - z_{2}}{z_{2} - z_{1}} = \frac{r_{3}}{r_{1}}$
and we know that $\frac{r_{3}}{r_{1}}$ is real
$\frac{z_{3} - z_{2}}{z_{2} - z_{1}}$ is purely real.

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Q. Prove that the points

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