Consider the locus arg(Z3−1Z3+1)=π2,Z∈C and choose correct option(s).

arg(Z3−1Z3+1)=π2Let Z3=w ⇒ w moves on a semicircle as shown below ∴ let w=eiθ,0≤θ≤π ⇒ z3=eiθ→z−eiθ/3,ei(θ+2π3),ei(θ−2π3)Locus of z is as shown below Shaded arc denotes the locus of z

Consider the locus arg(Z3−1Z3+1)=π2,Z∈C and choose correct option(s).

arg(Z3−1Z3+1)=π2Let Z3=w ⇒ w moves on a semicircle as shown below ∴ let w=eiθ,0≤θ≤π ⇒ z3=eiθ→z−eiθ/3,ei(θ+2π3),ei(θ−2π3)Locus of z is as shown below Shaded arc denotes the locus of z

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Consider the locus $\arg (\frac{Z^{3} - 1}{Z^{3} + 1}) = \frac{π}{2}, Z \in C$ and choose correct option(s).

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Total length of the locus of $Z$ is $2 π$

Total length of the locus of $Z$ is $π$

Length of the arc of the locus of $Z$ with $Re (Z) > 0$ and $Im (Z) < 0$ is $\frac{π}{6}$

Length of the arc of the locus of $Z$ with $Re (Z) > 0$ is $\frac{π}{2}$

answer is B, C, D.

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Detailed Solution

$\arg (\frac{Z^{3} - 1}{Z^{3} + 1}) = \frac{π}{2}$
Let $Z^{3} = w$
$\Rightarrow w$ moves on a semicircle as shown below
Question Image
$∴ let w = e^{i θ}, 0 \leq θ \leq π$
$\Rightarrow z^{3} = e^{i θ} \to z - e^{i θ / 3}, e^{i (\frac{θ + 2 π}{3})}, e^{i (\frac{θ - 2 π}{3})}$
Locus of $z$ is as shown below

Shaded arc denotes the locus of $z$