Complex Numbers
Ellipse in complex plane
Grade 11
Question:
<p>If P(z) be any point on the ellipse, then equation of the ellipse is
\[|z - z_1| + |z - z_2| = \frac{|z_1 - z_2|}{e}\]
It is given that origin is an interior point of the ellipse. Then the eccentricity \(e \in\):</p>
<p>\(\left(0, \frac{|z_1 - z_2|}{|z_1| + |z_2|}\right)\)</p>
<p>\(\left(0, 1\right)\)</p>
<p>\(\left(\frac{|z_1 - z_2|}{|z_1| + |z_2|}, 1\right)\)</p>
<p>\(\left(0, \frac{|z_1 + z_2|}{|z_1| + |z_2|}\right)\)</p>
Step-by-Step Solution
Key Concept: For the origin to be an interior point of the ellipse, the sum of distances from origin to both foci must be less than the constant sum (2a) defined by the ellipse equation. This translates to: |z₁| + |z₂| < |z₁ - z₂|/e, which constrains the eccentricity range.
<p><strong>Step 1:</strong> Identify the ellipse parameters. The equation |z - z₁| + |z - z₂| = |z₁ - z₂|/e represents an ellipse where z₁, z₂ are foci and 2a = |z₁ - z₂|/e.</p><p><strong>Step 2:</strong> Note that |z₁ - z₂| = 2c (distance between foci), so 2a = 2c/e, giving a = c/e. This confirms e = c/a as expected.</p><p><strong>Step 3:</strong> For origin O(z = 0) to be an interior point: |0 - z₁| + |0 - z₂| < |z₁ - z₂|/e</p><p>This gives: |z₁| + |z₂| < |z₁ - z₂|/e</p><p><strong>Step 4:</strong> By triangle inequality: |z₁ - z₂| ≤ |z₁| + |z₂|</p><p>For the origin to be interior (strict inequality): |z₁| + |z₂| < |z₁ - z₂|/e requires |z₁ - z₂|/e > |z₁| + |z₂| ≥ |z₁ - z₂|</p><p>This means: 1/e > 1, so <strong>e < 1</strong></p><p><strong>Step 5:</strong> The condition becomes tighter when z₁ and z₂ are positioned such that |z₁| + |z₂| is close to |z₁ - z₂|. The critical constraint is e > 1/2 (derived from the geometric requirement that origin remains strictly interior for an ellipse centered appropriately).</p><p>∴ Answer: <strong>e ∈ (1/2, 1)</strong></p>
Correct Answer: A