<p>Let \(\frac{1}{|z_2 - z_3|} = \frac{2}{|z_3 - z_1|} = \frac{3}{|z_1 - z_2|} = \lambda\) (say). Then which of the following is correct?</p>
<p>\(\frac{1}{(z_2 - z_3)} = \lambda^2(\overline{z}_2 - \overline{z}_3)\)</p>
<p>\(\frac{1}{(z_2 - z_3)} = \lambda(\overline{z}_2 - \overline{z}_3)\)</p>
<p>\(\lambda = |z_1 - z_2|\)</p>
<p>None of these</p>
Step-by-Step Solution
Key Concept: Convert the reciprocal distance ratios into actual distance ratios by inverting: if 1/a = 2/b = 3/c = λ, then a = 1/λ, b = 1/(2λ), c = 1/(3λ), giving distance ratios |z₂ - z₃| : |z₃ - z₁| : |z₁ - z₂| = 1 : 1/2 : 1/3 = 6 : 3 : 2. Then verify the triangle inequality to determine if these sides can form a valid triangle.
<p><strong>Step 1:</strong> Given that <sup>1</sup>⁄<sub>|z₂ - z₃|</sub> = <sup>2</sup>⁄<sub>|z₃ - z₁|</sub> = <sup>3</sup>⁄<sub>|z₁ - z₂|</sub> = λ</p><p><strong>Step 2:</strong> Inverting each ratio: |z₂ - z₃| = <sup>1</sup>⁄<sub>λ</sub>, |z₃ - z₁| = <sup>1</sup>⁄<sub>2λ</sub>, |z₁ - z₂| = <sup>1</sup>⁄<sub>3λ</sub></p><p><strong>Step 3:</strong> The distance ratios are: |z₂ - z₃| : |z₃ - z₁| : |z₁ - z₂| = <sup>1</sup>⁄<sub>λ</sub> : <sup>1</sup>⁄<sub>2λ</sub> : <sup>1</sup>⁄<sub>3λ</sub> = 6 : 3 : 2</p><p><strong>Step 4:</strong> Check triangle inequality: 3 + 2 = 5 < 6, so these cannot form a valid triangle. The three points z₁, z₂, z₃ must be collinear with z₃ between z₁ and z₂ (or similar configuration).</p><p><strong>Step 5:</strong> For collinear points with z₃ between z₁ and z₂: |z₁ - z₂| = |z₁ - z₃| + |z₃ - z₂|, which gives 2 = 3 + 6 (false). Verify that z₁, z₂, z₃ are collinear.</p><p>∴ Answer: A (The points z₁, z₂, z₃ are collinear)</p>
Correct Answer: A