The complex numbers z 1 - z 2 and z3 satisfying z 1 - z 3 z 2 - z 3 - 1- 3 i 2 are the vertices of a triangle which is-

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Properties of nth Root of a Complex Number

The complex n...

Question

The complex numbers $z_{1}, z_{2}$ and $z_{3}$ satisfying $\frac{z_{1} - z_{3}}{z_{2} - z_{3}} = \frac{1 - \sqrt{3} i}{2}$ are the vertices of a triangle which is:

of area zero

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right angled isosceles

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equilateral

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obtuse angled isosceles

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Solution

The correct option is C equilateral
If $z_{1}$ , $z_{2}$ and $z_{3}$ are the vertices of a triangle satisfying $\frac{z_{1} - z_{3}}{z_{2} - z_{3}}$ = $\frac{1 - i \sqrt{3}}{2}$ , then

$\frac{1 - i \sqrt{3}}{2}$ = $cos (π / 3) - i sin (π / 3)$

$\Rightarrow$ line segments made by $z_{1}$ , $z_{3}$ and $z_{2}$ , $z_{3}$ makes an angle of $60^{\circ}$ with each other $\Rightarrow$ given triangle is an equilateral triangle.

Hence, option C is correct.

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The complex numbers z1,z2 and z3 satisfying z1−z3z2−z3=1−√3i2 are the vertices of a triangle which is:

The complex numbers $z_{1}, z_{2}$ and $z_{3}$ satisfying $\frac{z_{1} - z_{3}}{z_{2} - z_{3}} = \frac{1 - \sqrt{3} i}{2}$ are the vertices of a triangle which is: