<p>If <span>\(|Z - 4| + |Z + 4| = 10\)</span>, then the difference between the maximum and the minimum values of <span>\(|Z|\)</span> is:</p>
Step-by-Step Solution
Key Concept: The equation |Z - 4| + |Z + 4| = 10 represents an ellipse with foci at 4 and -4 on the real axis, where the sum of distances from any point to the foci equals 10. We need to find the maximum and minimum values of |Z| (distance from origin) for points on this ellipse.
<p><strong>Step 1: Recognize the geometric locus.</strong> The equation |Z - 4| + |Z + 4| = 10 represents an ellipse in the complex plane with foci at F₁ = 4 and F₂ = -4 on the real axis. The sum of distances from any point Z to the two foci is constant (= 10).</p><p><strong>Step 2: Identify ellipse parameters.</strong> For an ellipse with foci at (±c, 0):<br/>• Distance between foci: 2c = 8, so c = 4<br/>• Sum of distances to foci: 2a = 10, so a = 5<br/>• Using b² = a² - c²: b² = 25 - 16 = 9, so b = 3</p><p><strong>Step 3: Write the ellipse equation.</strong> The ellipse equation in standard form is: x²/25 + y²/9 = 1, where Z = x + iy.</p><p><strong>Step 4: Find maximum value of |Z|.</strong> |Z|² = x² + y² must be maximized on the ellipse. From the ellipse equation: y² = 9(1 - x²/25)<br/>|Z|² = x² + 9(1 - x²/25) = x² + 9 - 9x²/25 = (16x²/25) + 9<br/>This is maximized when x² = 25 (at endpoints of major axis): |Z|²_max = 16 + 9 = 25, so |Z|_max = 5</p><p><strong>Step 5: Find minimum value of |Z|.</strong> |Z|² is minimized when x = 0 (at endpoints of minor axis):<br/>|Z|²_min = 0 + 9 = 9, so |Z|_min = 3</p><p><strong>Step 6: Calculate the difference.</strong> Difference = |Z|_max - |Z|_min = 5 - 3 = 2<br/>∴ Answer: A</p>
Correct Answer: A