<p>If <i>z</i> is a complex number in the argand plane, the equation <math>|z - 2| + |z + 2| = 8</math> represents</p>
Step-by-Step Solution
Key Concept: Recognize that the equation |z - 2| + |z + 2| = 8 represents the sum of distances from point z to two fixed points (foci at ±2), which defines an ellipse when this sum is constant and greater than the distance between the foci.
<p><strong>Step 1:</strong> Interpret the equation in terms of distances. Let z = x + iy. Then |z - 2| represents the distance from z to point F₁ = (2, 0) and |z + 2| represents the distance from z to point F₂ = (-2, 0).</p><p><strong>Step 2:</strong> The equation becomes: distance from z to F₁ + distance from z to F₂ = 8, or |PF₁| + |PF₂| = 8 (where P represents point z).</p><p><strong>Step 3:</strong> Check the condition for an ellipse. For an ellipse, the sum of distances from any point on the curve to two fixed points (foci) equals a constant, and this constant must be greater than the distance between the foci. Here: distance between foci = |2 - (-2)| = 4, and the given sum = 8. Since 8 > 4, the condition is satisfied.</p><p><strong>Step 4:</strong> Verify this is not a degenerate case. Since 8 ≠ 4 (which would give a line segment), we have a proper ellipse with foci at F₁(2, 0) and F₂(-2, 0), where 2a = 8, so a = 4, and c = 2 (distance from center to focus).</p><p><strong>∴ Answer:</strong> B</p>
Correct Answer: B