Paragraph: Let $S=S_{1} \cap S_{2} \cap S_{3}$, where $S_{1}=\{z \in \mathbb{C}:|z| 0\right\}$ and…

Paragraph: Let $S=S_{1} \cap S_{2} \cap S_{3}$, where $S_{1}=\{z \in \mathbb{C}:|z|<4\}, \quad S_{2}=\left\{z \in \mathbb{C}: \operatorname{Im}\left[\frac{z-1+\sqrt{3} i}{1-\sqrt{3} i}\right]>0\right\}$ and $S_{3}=\{z \in \mathbb{C}: \operatorname{Re} z>0\}$.
Question: $\min _{z \in S}|1-3 i-\mathrm{z}|=$
  1. 2 - 3 2
  2. 2 + 3 2
  3. 3 - 3 2
  4. 3 + 3 2

Solution

min z ∈ s 1 - 3 i - z perpendicular length of point (1, -3) from line 3 x +  y = 0
3 - 3 3 + 1 = 3 - 3 2 = 3 - 3 2

Asked in: JEE Advanced 2013 (Paper 2)

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