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Find the number of complex numbers z satisfying |z-1|=|z+1|=|z

Question

Find the number of complex numbers $z$ satisfying $∣ z - 1 ∣ = ∣ z + 1 ∣ = ∣ z - i ∣$

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Find the number of complex numbers

z

satisfying

∣ z - 1 ∣ = ∣ z + 1 ∣ = ∣ z - i ∣

12 mins ago

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Text solutionVerified

Let

z = x + i y

∣ z - 1 = ∣ x + i y - 1∣ = (x - 1)^{2} + y^{2}

∣ z + 1∣ = ∣ x + i y + 1∣ = (x + 1)^{2} + y^{2}

∣ z - i ∣ = ∣ x + i y - i ∣ = x^{2} + (y - 1)^{2}

Given

∣ z - 1∣ = ∣ z + 1∣ = ∣ z - i ∣

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

Squaring , we get

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

Now considering,

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

\Rightarrow x - 1 = \pm (x + 1)

\Rightarrow x-1 = x+1, \ \ \ \ x-1 = -x-1

Considering

x - 1 = - x - 1

\Rightarrow 2 x = 0

∴ x = 0

and now considering

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

put

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

∴ y = 0

∴ z = 0

∴

there is only one complex number.

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Question Text	Find the number of complex numbers $z$ satisfying $∣ z - 1 ∣ = ∣ z + 1 ∣ = ∣ z - i ∣$
Answer Type	Text solution:1
Upvotes	150