# On which axis does the complex number z = x + iy, which satisfies the equation ∣z − 5i| / |z+5i∣=1 lies?

A complex number is needed to be converted into its real values to solve it further.

## Answer: The complex number z lies on the x-axis.

We may make use of the __modular properties__ of a complex number.

**Explanation:**

Given, ∣ z − 5i | / | z + 5i ∣ = 1

==> ∣ z − 5i | = | z + 5i ∣ ----(1)

Let the complex number Z be written in real and complex part form.

Therefore, z = x + iy, where x is the real part and y is the imaginary part.

Squaring the equation two on both sides, and replacing z with x + iy, we get

∣z − 5i |^{2} = | z + 5i ∣^{2}

==> | x + iy - 5i |^{2 }= |x + iy + 5i |^{2}

==>^{ }| x + i(y - 5) |^{2 }= ^{ }| x + i(y + 5) |^{2}

==> x^{2} + (y - 5)^{2} = x^{2} + (y + 5)^{2}

==> x^{2} + y^{2} + 25 -10y = x^{2} + y^{2} + 25 + 10y

Thus, on canceling even terms from both sides, we get

==> 20 y = 0 or y = 0

### Hence the complex number z lies on the x-axis.

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