Let S1 = {z ∈ C:|z| ≤ 5}, S2 = {z∈C : Im(z+1−√3i / 1−√3i) ≥ 0} and S3 = {z ∈ C : Re(z) ≥ 0}.

Question

Let \(S_{1}=\{z \in C:|z| \leq 5\}\), \(S_{2}=\left\{z \in C: \operatorname{Im}\left(\frac{z+1-\sqrt{3} i}{1-\sqrt{3} i}\right) \geq 0\right\}\) and\(S_{3}=\{z \in C: \operatorname{Re}(z) \geq 0\}\). Then the area of region \(\mathrm{S}_{1} \cap \mathrm{S}_{2} \cap \mathrm{S}_{3}\) is

(1) \(\frac{125 \pi}{6}\)

(2) \(\frac{125 \pi}{24}\)

(3) \(\frac{125 \pi}{4}\)

(4) \(\frac{125 \pi}{12}\)

Answer 1

Correct option is (4) \(\frac{125 \pi}{12}\)

\(\mathrm S_1 \to\) area inside circle with radius 5

\(\mathrm{S}_2 =\mathrm{I}_{\mathrm{m}}\left(\frac{(\mathrm{x}+1)+\mathrm{i}(\mathrm{y}-\sqrt{3})}{1-\sqrt{3} \mathrm{i}} \times \frac{1+\sqrt{3} \mathrm{i}}{1+\sqrt{3} \mathrm{i}}\right)\)

\(=\mathrm{I}_{\mathrm{m}}\left(\frac{[(\mathrm{x}+1)+\mathrm{i}(\mathrm{y}-\sqrt{3})](1+\sqrt{3} \mathrm{i})}{1+3}\right)\)

\(=\sqrt{3} \mathrm{x}+\sqrt{3}+\mathrm{y}-\sqrt{3} \geq 0\)

\(= \sqrt{3} \mathrm{x}+\mathrm{y} \geq 0\)

\( \mathrm{S}_3=\mathrm{x} \geq 0\)

Area of half circle - area of arc AB

\(=\frac{25 \pi}{2}-\frac{1}{2} \times 25 \times \frac{\pi}{6}\)

\(=\frac{25 \pi}{2}-\frac{25 \pi}{12} \)

\(=\frac{125 \pi}{12}\)