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Equation of Circle in Complex Form

If z=4-3i is ...

Question

If $z = 4 - 3 i$ is rotated by $180^{\circ}$ in the clockwise direction about origin and stretched three times of its original length, then the new complex number is

12 + 9 i

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12 - 9 i

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- 12 - 9 i

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- 12 + 9 i

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Solution

The correct option is D $- 12 + 9 i$
Given complex number $z = 4 - 3 i$
Let the new rotated complex number be $z_{1}$ , then
$| z_{1} | = 3 | z |$
Also,
$\frac{z_{1}}{z} = \frac{| z_{1} |}{| z |} e^{- i π} \Rightarrow \frac{z_{1}}{z} = - 3 ∴ z_{1} = - 12 + 9 i$

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[Kerala (Engg.) 2005]

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Equation of Circle in Complex Form

Standard XII Mathematics

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If z=4−3i is rotated by 180∘ in the clockwise direction about origin and stretched three times of its original length, then the new complex number is

The correct option is D −12+9iGiven complex number z=4−3i Let the new rotated complex number be z1, then |z1|=3|z| Also, z1z=|z1||z|e−iπ⇒z1z=−3∴z1=−12+9i

If $z = 4 - 3 i$ is rotated by $180^{\circ}$ in the clockwise direction about origin and stretched three times of its original length, then the new complex number is

The correct option is D $- 12 + 9 i$
Given complex number $z = 4 - 3 i$
Let the new rotated complex number be $z_{1}$ , then
$| z_{1} | = 3 | z |$
Also,
$\frac{z_{1}}{z} = \frac{| z_{1} |}{| z |} e^{- i π} \Rightarrow \frac{z_{1}}{z} = - 3 ∴ z_{1} = - 12 + 9 i$