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.
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.