<p>The locus of the centre of a circle which touches the circle \(|z-z_1|=a\) and \(|z-z_2|=b\) externally (\(z\), \(z_1\) and \(z_2\) are complex numbers) may be</p>
Step-by-Step Solution
Key Concept: When a circle with centre z and radius r touches two fixed circles externally, the centre z must satisfy |z - z₁| - r = a and |z - z₂| - r = b, which eliminates r to give ||z - z₁| - |z - z₂|| = |a - b|, the definition of a hyperbola (or degenerate cases).
<p><strong>Step 1:</strong> Let the variable circle have centre z and radius r, touching circle |z - z₁| = a externally and |z - z₂| = b externally.</p><p><strong>Step 2:</strong> For external tangency: |z - z₁| = a + r and |z - z₂| = b + r</p><p><strong>Step 3:</strong> Subtracting these equations: |z - z₁| - |z - z₂| = (a + r) - (b + r) = a - b</p><p><strong>Step 4:</strong> This gives ||z - z₁| - |z - z₂|| = |a - b|, which is the equation of a hyperbola with foci at z₁ and z₂.</p><p><strong>Step 5:</strong> If a = b, the locus degenerates to the perpendicular bisector of the segment joining z₁ and z₂ (a line, special case of hyperbola).</p><p>∴ The locus is a hyperbola (or line when a = b)</p>
Correct Answer: A