Z is a complex number such that Z≤2 and -π3≤ampZ≤π3. The area of the region formed by locus of Z is (in sq. units)

TS EAMCET · Maths · Complex Number

$Z$ is a complex number such that $|Z| \leq 2$ and $- \frac{π}{3} \leq amp Z \leq \frac{π}{3}$ . The area of the region formed by locus of $Z$ is (in sq. units)

A $\frac{2 π}{3}$
B $\frac{π}{3}$
C $\frac{4 π}{3}$
D $\frac{8 π}{3}$

Verified Solution

Answer & Solution

Correct Answer

(C) $\frac{4 π}{3}$

Step-by-step Solution

Detailed explanation

Given Z≤2 and -π3≤ampZ≤π3. This represents a sector of a circle of whose radius is less than or equal to 2 units with the central angle equal to 2π3. From the above diagram, the locus of Z will form a sector OABO. Thus, the area of the region…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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