The complex numbers z 1- z 2- z 3 satisfying z 1- z 3- z 2- z 3-1- i -3-2 are the vertices of a vertices of a triangle which isA- right angled isoscelesB- of area zeroC- obtuse angled isoscelesD- equilateral

In equilateral triangle arg z_1 z_3/z_2 z_3= π/3 and | z_1 z_3|=| z_2 z_3| Solution :arg z_1 z_3/z_2 z_3= π/3 (z_1 z_3)=(z_2 z_3)e^ π/3i (z_1 z_3)/(z_2 z_3) ...

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Byju's Answer

Standard XII

Mathematics

Properties of Conjugate of a Complex Number

The complex n...

Question

The complex numbers $z_{1}, z_{2}, z_{3}$ satisfying $\frac{z_{1} - z_{3}}{z_{2} - z_{3}} = \frac{1 - i \sqrt{3}}{2}$ are the vertices of a 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
In equilateral triangle arg $\frac{z_{1} - z_{3}}{z_{2} - z_{3}} = - \frac{π}{3}$

and $∣ z_{1} - z_{3} ∣=∣ z_{2} - z_{3} ∣$

Solution :arg $\frac{z_{1} - z_{3}}{z_{2} - z_{3}} = - \frac{π}{3}$

$(z_{1} - z_{3}) = (z_{2} - z_{3}) e^{-} \frac{π}{3 i}$

$\frac{(z_{1} - z_{3})}{(z_{2} - z_{3})} = c o s \frac{π}{3} - i s i n \frac{π}{3 i} = \frac{1 - i \sqrt{3}}{2}$
$∣ \frac{z_{1} - z_{3}}{z_{2} - z_{3}} ∣=∣ \frac{1 - i \sqrt{3}}{2} ∣= 1$

$∣ z_{1} - z_{3} ∣=∣ z_{2} - z_{3} ∣$
Triangle is equilateral

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