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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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Similar questions

z_{1}, z_{2}, z_{3}

are imaginary numbers such that

| z_{1} | = | z_{2} | = | z_{3} | = ∣ ∣ ∣ \begin{matrix} \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} \end{matrix} ∣ ∣ ∣ = 1

| z_{1} + z_{2} + z_{3} |

Q. Prove that the points

z_{1}, z_{2}, z_{3}, z_{4}

taken in order are concyclic if and only if

\frac{(z_{3} - z_{1}) (z_{4} - z_{2})}{(z_{3} - z_{2}) (z_{4} - z_{1})}

is purely real.

z_{1}, z_{2}, z_{3}

be complex number such that

| z_{1} | = | z_{2} | = | z_{3} | = ∣ ∣ \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} ∣ ∣

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