Q.Where does $z$ lie, if $\left|\dfrac{z-5i}{z+5i}\right|=1$.

The problem asks for the locus of $z$ given a condition involving magnitudes of complex numbers. The key idea here is the geometric interpretation of the modulus of a complex number and its application in complex number arithmetic.

The given condition is $\left|\dfrac{z-5i}{z+5i}\right|=1$.

Using the property that $|z_1/z_2| = |z_1|/|z_2|$, we can rewrite this as:

This implies $|z-5i| = |z+5i|$.

Therefore, the equation $|z-5i| = |z-(-5i)|$ means that the distance from $z$ to $5i$ is equal to the distance from $z$ to $-5i$. …

When working with complex numbers, it is often helpful to think about their geometric interpretation. The expression $|z_1 - z_2|$ represents the distance between the complex numbers $z_1$ and $z_2$ in the complex plane. This fundamental idea is key to understanding the given problem.

The equation we have is $\left|\dfrac{z-5i}{z+5i}\right|=1$. Let's break down what this means.

1. Interpret the equation geometrically. The property of moduli states that for any two complex numbers $w_1$ and $w_2$, $\left|\dfrac{w_1}{w_2}\right| = \dfrac{|w_1|}{|w_2|}$. Applying this to our equation:

Multiplying both sides by $|z+5i|$ (which cannot be zero, otherwise $z = -5i$, making the denominator zero in the original expression, which is undefined), we get:

This can be rewritten as:

Now, let's identify the points involved.
*   $|z-5i|$ represents the distance between the complex number $z$ and the fixed complex number $5i$. Let's call $A = 5i$.
*   $|z-(-5i)|$ represents the distance between the complex number $z$ and the fixed complex number $-5i$. Let's call $B = -5i$.

So, the equation $|z-5i| = |z-(-5i)|$ means that the distance from $z$ to $5i$ is equal to the distance from $z$ to $-5i$. In other words, $z$ is equidistant from the points $A(0, 5)$ and $B(0, -5)$ in the complex plane.

2. Recall the geometric locus.

The locus of all points that are equidistant from two fixed points is the perpendicular bisector of the line segment joining those two points.

3. Find the perpendicular bisector.

   • The two fixed points are $A = 5i$ (which corresponds to the Cartesian coordinate $(0, 5)$) and $B = -5i$ (which corresponds to $(0, -5)$).
   • The line segment joining $A$ and $B$ lies entirely on the imaginary axis. It is a vertical segment from $(0, -5)$ to $(0, 5)$.
   • The midpoint of this segment is $\dfrac{5i + (-5i)}{2} = \dfrac{0}{2} = 0$. This corresponds to the origin $(0, 0)$.
   • Since the segment $AB$ is vertical, its perpendicular bisector must be a horizontal line.
   • This horizontal line must pass through the midpoint, which is the origin $(0, 0)$.
   • The horizontal line passing through the origin is the real axis (where the imaginary part of $z$ is zero).

   Therefore, $z$ must lie on the real axis. …