If S = { z ? C : | z - i | = | z + i | = | z - 1 | } then n ( S ) is: [2024]

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Solution

Q.
If $S = {z \in C :$ $| z - i |$ $=$ $| z + i |$ $=$ $| z - 1 |$ $}$ then $n (S)$ is: [2024]

1 1

2 0

3 3

4 2

Ans.
(1)

Given, $| z - i |$ $=$ $| z + i |$ $=$ $| z - 1 |$ ...(i)

Putting $z = x + i y$ in (i), we get $| x + i y - i |$ $=$ $| x + i y + i |$ $=$ $| x + i y - 1 |$

Now, $| x + i (y - 1) |$ $=$ $| x - 1 + i y |$

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

$\Rightarrow x^{2} + y^{2} + 1 - 2 y = x^{2} + 1 - 2 x + y^{2} \Rightarrow 2 x - 2 y = 0$

$\Rightarrow x - y = 0$ ...(ii)

Again, $| x + i (y + 1) |$ $=$ $| x - 1 + i y |$

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

$\Rightarrow x^{2} + y^{2} + 1 + 2 y = x^{2} + 1 - 2 x + y^{2} \Rightarrow 2 x + 2 y = 0$

$\Rightarrow x + y = 0$ ...(iii)

From (ii) and (iii), we get $x = 0$ and $y = 0$

Thus, $z = 0 + i 0 ∴ n (S) = 1$

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