Find the locus of the points representing the complex number `z` for which`|z+5|^2=|z-5|^2=10.`

Explanation

To find the locus of the points representing the complex number \( z \) for which \( |z + 5|^2 = |z - 5|^2 = 10 \), we can follow these steps:

### Step 1: Set up the equations
We know that \( |z + 5|^2 = |z - 5|^2 \). This means the distance from \( z \) to \( -5 \) is equal to the distance from \( z \) to \( 5 \).

### Step 2: Use the definition of modulus
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. Then we can express the moduli as:
\[
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