<p>Find the locus of a complex number <i>z</i> = <i>x</i> + <i>iy</i>, which satisfy the equation <span>\(\frac{z - 5i}{z + 5i} = 1\)</span>.</p>
Step-by-Step Solution
Key Concept: The modulus of a quotient equals the quotient of moduli. Setting equal moduli gives the locus equation.
<p><strong>Step 1:</strong> Put <i>z</i> = <i>x</i> + <i>iy</i> in the given equation.</p><p>$\frac{x + iy - 5i}{x + iy + 5i} = 1$</p><p><strong>Step 2:</strong> Simplify:</p><p>$\frac{x + i(y - 5)}{x + i(y + 5)} = 1$</p><p><strong>Step 3:</strong> Since <span>$\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$</span>, we have:</p><p>$|x + i(y - 5)| = |x + i(y + 5)|$</p><p><strong>Step 4:</strong> Taking modulus:</p><p>$x^2 + (y - 5)^2 = x^2 + (y + 5)^2$</p><p><strong>Step 5:</strong> Expand:</p><p>$x^2 + y^2 - 10y + 25 = x^2 + y^2 + 25 + 10y$</p><p><strong>Step 6:</strong> Simplify:</p><p>$-10y = 10y$</p><p>$20y = 0 \Rightarrow y = 0$</p><p>Hence, it lies on the X-axis. ∴ Answer is (a).</p>
Correct Answer: A