Parabola
Focal Chord Properties
Grade 11
Question:
<p>If \(A = (at^2, 2at)\), \(B = \left(\frac{a}{t^2}, -\frac{2a}{t}\right)\) and \(S = (a, 0)\), prove that \(2a\) is the harmonic mean of \(SA\) and \(SB\).</p>
Step-by-Step Solution
Key Concept: Use the focal chord property: if A and B lie on a parabola y² = 4ax with S as focus, then 1/SA + 1/SB = 1/a. The harmonic mean formula 2/HM = 1/SA + 1/SB directly gives HM = 2a.
<p><strong>Step 1:</strong> Identify that S = (a, 0) is the focus of parabola y² = 4ax. Points A = (at², 2at) and B = (a/t², -2a/t) both satisfy y² = 4ax (verify: (2at)² = 4a·at² ✓).</p><p><strong>Step 2:</strong> Calculate SA using distance formula:</p><p>SA² = (at² - a)² + (2at - 0)² = a²(t² - 1)² + 4a²t² = a²(t⁴ - 2t² + 1 + 4t²) = a²(t² + 1)²</p><p>∴ SA = a(t² + 1)</p><p><strong>Step 3:</strong> Calculate SB similarly:</p><p>SB² = (a/t² - a)² + (-2a/t)² = a²(1/t² - 1)² + 4a²/t² = a²[(1 - t²)²/t⁴ + 4/t²] = a²(1 + t²)²/t⁴</p><p>∴ SB = a(1 + t²)/t²</p><p><strong>Step 4:</strong> Use harmonic mean formula. If H is harmonic mean of SA and SB:</p><p>1/SA + 1/SB = 1/[a(t² + 1)] + t²/[a(t² + 1)] = (1 + t²)/[a(t² + 1)] = 1/a</p><p><strong>Step 5:</strong> Therefore: 2/H = 1/a, which gives H = 2a</p><p>∴ <strong>Proven: 2a is the harmonic mean of SA and SB</strong></p>
Correct Answer: Proven: 2a is harmonic mean of SA and SB