Complex Numbers
Modulus and Argument
Grade 11
Question:
<p>If \(z\) is a complex number such that \(|z| \geq 2\), then the minimum value of \(\left|z + \dfrac{1}{2}\right|\)</p>
<p>is equal to \(\dfrac{5}{2}\)</p>
<p>lies in the interval \((1, 2)\)</p>
<p>is strictly greater than \(\dfrac{5}{2}\)</p>
<p>is strictly greater than \(\dfrac{3}{2}\) but less than \(\dfrac{5}{2}\)</p>
Step-by-Step Solution
Key Concept: The minimum distance from a point z on or outside the circle |z|=2 to the point -1/2 occurs along the line connecting the origin to -1/2. Since -1/2 lies inside the circle |z|=2, the minimum is achieved at the boundary point closest to -1/2.
<p><strong>Step 1:</strong> Recognize that we need to minimize |z + 1/2| subject to the constraint |z| ≥ 2.</p><p><strong>Step 2:</strong> Geometrically, z + 1/2 represents the distance from the point z to the point -1/2. The set of z with |z| ≥ 2 represents all points outside or on the circle of radius 2 centered at origin.</p><p><strong>Step 3:</strong> Since -1/2 is inside the circle (as |-1/2| = 1/2 < 2), the closest point on the boundary |z| = 2 to -1/2 lies on the line segment from O to -1/2, at distance 2 from origin in the direction of -1/2.</p><p><strong>Step 4:</strong> The closest point is z = -1 (which satisfies |z| = 1·|−1/2|/|−1/2| × 2 = 2 in direction of -1/2).</p><p><strong>Step 5:</strong> Calculate: |z + 1/2| = |-1 + 1/2| = |-1/2| = 1/2. Wait, recalculate: the point is z = -2·(1/2)/|1/2| = -2, so |z + 1/2| = |-2 + 1/2| = |-3/2| = <strong>3/2</strong>.</p><p>∴ Answer: A (minimum value = 3/2)</p>
Correct Answer: A