<p>If \(|z - 1| + |z + 3| \leq 8\), then find the range of values of \(|z - 4|\).</p>
Step-by-Step Solution
Key Concept: The constraint |z - 1| + |z + 3| ≤ 8 defines an ellipse with foci at 1 and -3. Use the triangle inequality and the geometric property that |z - 4| ranges from the minimum distance from 4 to the ellipse to the maximum distance.
<p><strong>Step 1:</strong> The constraint |z - 1| + |z + 3| ≤ 8 represents the interior and boundary of an ellipse with foci F₁ = 1 and F₂ = -3. The distance between foci is |1 - (-3)| = 4, so 2c = 4, giving c = 2. Since 2a = 8, we have a = 4.</p><p><strong>Step 2:</strong> The center of the ellipse is at the midpoint: (-1, 0). We have b² = a² - c² = 16 - 4 = 12, so b = 2√3.</p><p><strong>Step 3:</strong> To find the range of |z - 4|, we need distances from point 4 (on the real axis) to all points in the elliptical region. The point 4 lies on the major axis at distance |4 - (-1)| = 5 from the center.</p><p><strong>Step 4:</strong> The right vertex of the ellipse is at -1 + 4 = 3. The minimum distance from 4 to the ellipse occurs at the right vertex: |4 - 3| = 1.</p><p><strong>Step 5:</strong> The left vertex is at -1 - 4 = -5. The maximum distance from 4 to the ellipse occurs at the left vertex: |4 - (-5)| = 9.</p><p><strong>Step 6:</strong> Since the region includes the interior of the ellipse and all points from the boundary satisfy these extreme values, the range of |z - 4| is continuous from minimum to maximum.</p><p>∴ Answer: <strong>1 ≤ |z - 4| ≤ 9</strong></p>
Correct Answer: 1 ≤ |z - 4| ≤ 9