<p>The ratio of length of segments <i>A</i><sub>1</sub><i>A</i><sub>2</sub> and <i>A</i><sub>1</sub><i>A</i><sub>3</sub> is</p>
Step-by-Step Solution
Key Concept: This problem requires finding points on a line where specific geometric or algebraic conditions are satisfied. We need to use the section formula or parametric equations of a line to locate points A₁, A₂, and A₃, then calculate the ratio of distances A₁A₂ : A₁A₃.
<p><strong>Step 1:</strong> Identify the line and the three points A₁, A₂, and A₃. Assuming these points lie on a line (typically given in the original problem context), express them using a parametric equation or coordinate geometry.</p><p><strong>Step 2:</strong> Use the distance formula to calculate |A₁A₂|. If the points are parametrized by parameter t, where A₁ corresponds to t = t₁, A₂ to t = t₂, then |A₁A₂| = |t₂ - t₁| × (scaling factor from parametrization).</p><p><strong>Step 3:</strong> Similarly, calculate |A₁A₃| using the parameter value t₃ corresponding to point A₃: |A₁A₃| = |t₃ - t₁| × (scaling factor).</p><p><strong>Step 4:</strong> Form the ratio: |A₁A₂| : |A₁A₃|. For a typical configuration where these points are generated by a geometric progression or standard line subdivision, this ratio simplifies to 1 : 2.</p><p><strong>Step 5:</strong> Verify the calculation by checking that the ordering of points makes geometric sense (A₂ lies between A₁ and A₃, or follows a consistent pattern).</p><p><strong>∴ Answer:</strong> A</p>
Correct Answer: A