The complex numbers z 1 , z 2 and z 3 satisfying z 1 - z 3 z 2 - z 3 = 1 - i 3 2 are the vertices of a triangle which is [2001]

Question

The complex numbers $z_{1}, z_{2}$ and $z_{3}$ satisfying $\frac{z_{1} - z_{3}}{z_{2} - z_{3}} = \frac{1 - i \sqrt{3}}{2}$ are the vertices of a triangle which is [2001]

Smart Achievers · Accepted Answer

(3)

$\frac{z_{1} - z_{3}}{z_{2} - z_{3}} = \frac{1 - i \sqrt{3}}{2}$

$\Rightarrow \arg (\frac{z_{1} - z_{3}}{z_{2} - z_{3}}) = \arg (\frac{1 - i \sqrt{3}}{2})$

$\Rightarrow \arg (\cos (- \frac{π}{3}) + i \sin (- \frac{π}{3}))$

$\Rightarrow angle between (z_{1} - z_{3}) and (z_{2} - z_{3}) is 60 ° .$

$and$ $| \frac{z_{1} - z_{3}}{z_{2} - z_{3}} |$ $=$ $| \frac{1 - i \sqrt{3}}{2} |$

$\Rightarrow$ $| \frac{z_{1} - z_{3}}{z_{2} - z_{3}} |$ $= 1 \Rightarrow | z_{1} - z_{3} | = | z_{2} - z_{3} | (Imp Step)$

$\Rightarrow The ∆ with vertices z_{1}, z_{2}, z_{3} is isosceles with vertical angle 60 ° .$

$\Rightarrow Hence rest of the two angles should also be 60 ° each.$

$\Rightarrow Required triangle is an equilateral triangle.$

Solution

Q.
The complex numbers $z_{1}, z_{2}$ and $z_{3}$ satisfying $\frac{z_{1} - z_{3}}{z_{2} - z_{3}} = \frac{1 - i \sqrt{3}}{2}$ are the vertices of a triangle which is [2001]

1 of area zero

2 right-angled isosceles

3 equilateral

4 obtuse-angled isosceles

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