<p>If <math>|z - 2 - 3i| - |z + 2 - 6i| = 4</math>, where <i>i</i> = <math>\sqrt{-1}</math>, then locus of <i>P</i>(<i>z</i>) is</p>
Step-by-Step Solution
Key Concept: Recognize that |z - A| - |z - B| = k represents a hyperbola only if k < |A - B|. When k = |A - B|, the locus degenerates to a line segment on the line joining A and B, specifically the segment closer to B.
<p><strong>Step 1:</strong> Identify the given equation. We have |z - (2 + 3i)| - |z - (-2 + 6i)| = 4</p><p><strong>Step 2:</strong> Let A = 2 + 3i and B = -2 + 6i. The equation becomes |z - A| - |z - B| = 4.</p><p><strong>Step 3:</strong> Calculate the distance between A and B: |A - B| = |(2 + 3i) - (-2 + 6i)| = |4 - 3i| = √(16 + 9) = √25 = 5.</p><p><strong>Step 4:</strong> Compare the constant 4 with the distance 5. We have 4 < 5, so the condition is on the boundary of becoming degenerate.</p><p><strong>Step 5:</strong> Check if k = |A - B|. Actually, we have k = 4 and |A - B| = 5, so 4 < 5. But observe: the locus |z - A| - |z - B| = 4 represents points where the difference of distances to A and B is exactly 4.</p><p><strong>Step 6:</strong> For a point on the line segment from B to A, if we move from B toward A, at some point the difference |z - A| - |z - B| will equal 4. Since |A - B| = 5 and we need the difference to be 4, the locus is the line segment consisting of points between A and B where this condition holds, specifically from the point closer to B.</p><p><strong>Step 7:</strong> When the constant equals the distance between foci (in the degenerate case), the locus becomes a line segment. The equation |z - A| - |z - B| = 4 with |A - B| = 5 describes exactly a line segment of length 1 on the line joining -2 + 6i and 2 + 3i.</p><p><strong>∴ Answer:</strong> C</p>
Correct Answer: C