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Q.

Given a parallelogram having vertices as $z_{1}, z_{2}, z_{3}, z_{4}$ as shown below.

If $z_{3} = α z_{1} + β z_{2}$ $(where α, β are complex numbers)$ , then which of the following is/are CORRECT?

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a

$\frac{| α |}{| β |} = 2$

b

$a m p (α) + a m p (β) = \frac{π}{6}$

c

Area of triangle formed by origin with $α a n d β$ is $\frac{\sqrt{3}}{2}$ Sq. units

d

Circumradius of triangle formed by origin with $α a n d β$ is $\sqrt{13}$ units

answer is B, C, D.

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Detailed Solution

Question Image

Using rotation, we get $\frac{z_{3} - z_{2}}{z_{1} - z_{2}} = 2 e^{- i \frac{π}{3}}$ $\Rightarrow z_{3} = z_{1} (2 e^{- i \frac{π}{3}}) + z_{2} (1 - 2 e^{- i \frac{π}{3}})$
So, we get $α = 2 e^{- i \frac{π}{3}}$ and $β = 1 - 2 e^{- i \frac{π}{3}} = \sqrt{3} i$

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Given a parallelogram having vertices as z1,z2,z3,z4 as shown below.If z3=αz1+βz2(where α,β are complex numbers), then which of the following is/are CORRECT?