<p>If <em>z</em> is any complex number such that <br>\(|3z - 2| + |3z + 2| = 4\), then identify the locus of <em>z</em>.</p>
<p>A circle with centre at origin</p>
<p>A line segment joining \(\left(\frac{2}{3}, 0\right)\) and \(\left(-\frac{2}{3}, 0\right)\)</p>
<p>A parabola</p>
<p>An ellipse</p>
Step-by-Step Solution
Key Concept: Recognize that |3z - 2| + |3z + 2| = 4 represents the sum of distances from 3z to two fixed points (2 and -2). This is the definition of an ellipse, where the sum of distances to two foci is constant. Divide by 3 to find the locus of z.
<p><strong>Step 1:</strong> Rewrite the equation in terms of 3z. We have |3z - 2| + |3z + 2| = 4.</p><p><strong>Step 2:</strong> Recognize this as |3z - 2| + |3z - (-2)| = 4. This represents the sum of distances from point 3z to the fixed points F₁ = 2 and F₂ = -2 being constant (equals 4).</p><p><strong>Step 3:</strong> By the definition of an ellipse, the locus of 3z is an ellipse with foci at ±2 and 2a = 4, so a = 2.</p><p><strong>Step 4:</strong> The distance between foci is 2c = 4, so c = 2. But since a = 2 and c = 2, we have a = c, which means b² = a² - c² = 0. This indicates the ellipse degenerates to a line segment (the major axis).</p><p><strong>Step 5:</strong> For 3z: the locus is the line segment from -2 to 2 on the real axis. Dividing by 3: z lies on the line segment from -2/3 to 2/3 on the real axis, i.e., the real axis with |z| ≤ 2/3, or equivalently Re(z) ∈ [-2/3, 2/3] and Im(z) = 0.</p><p>∴ The locus of z is the line segment on the real axis: -2/3 ≤ Re(z) ≤ 2/3, Im(z) = 0</p>
Correct Answer: B