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

==> x² + (y - 5)²  = x² + (y + 5)²

==> x²  + y² + 25 -10y = x² + y² + 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.