Let denote the set of all real numbers. Let z 1 = 1 + 2 i and z 2 = 3 i be two complex numbers, where i = - 1 . Let S = { ( x , y ): | x + i y - z 1 | = 2 | x + i y - z 2 | } . Then which of the following statements is (are) TRUE? [2025]

Question

Let $ℝ$ denote the set of all real numbers. Let $z_{1} = 1 + 2 i$ and $z_{2} = 3 i$ be two complex numbers, where $i = \sqrt{- 1}$ .

Let $S = {(x, y) \in ℝ \times ℝ : | x + i y - z_{1} | = 2 | x + i y - z_{2} |} .$ Then which of the following statements is (are) TRUE? [2025]

Smart Achievers · Accepted Answer

(1, 4)

$We have, | x + i y - 1 - 2 i | = 2 | x + i y - 3 i |$

$\Rightarrow {(x - 1)}^{2} + {(y - 2)}^{2} = 4 (x^{2} + {(y - 3)}^{2})$

$\Rightarrow 3 x^{2} + 3 y^{2} + 2 x - 20 y + 31 = 0$

$\Rightarrow x^{2} + y^{2} + \frac{2 x}{3} - \frac{20 y}{3} + \frac{31}{3} = 0$

$∴$ $S is a circle with centre (- \frac{1}{3}, \frac{10}{3}) and radius$

$= \sqrt{\frac{1}{9} + \frac{100}{9} - \frac{31}{3}} = \sqrt{\frac{8}{9}} = \frac{2 \sqrt{2}}{3}$

Solution

Q.
Let $ℝ$ denote the set of all real numbers. Let $z_{1} = 1 + 2 i$ and $z_{2} = 3 i$ be two complex numbers, where $i = \sqrt{- 1}$ .

Let $S = {(x, y) \in ℝ \times ℝ : | x + i y - z_{1} | = 2 | x + i y - z_{2} |} .$ Then which of the following statements is (are) TRUE [2025]

1 $S$ is a circle with centre $(- \frac{1}{3}, \frac{10}{3})$

2 $S$ is a circle with centre $(\frac{1}{3}, \frac{8}{3})$

3 $S$ is a circle with radius $\frac{\sqrt{2}}{3}$

4 $S$ is a circle with radius $\frac{2 \sqrt{2}}{3}$

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Solution

Q. Let ℝ denote the set of all real numbers. Let z1=1+2i and z2=3i be two complex numbers, where i=-1. Let S={(x,y)∈ℝ×ℝ:|x+iy-z1|=2|x+iy-z2|}. Then which of the following statements is (are) TRUE [2025]

1 S is a circle with centre (-13,103)

2 S is a circle with centre (13,83)

3 S is a circle with radius 23

4 S is a circle with radius 223