Parabola
Focal Chords
Grade 11
Question:
<p>If ∠ORP = π/2 = ∠ORQ where P(at₁², 2at₁) and Q(at₂², 2at₂) are points on the parabola y² = 4ax, and the slope of PQ is 2/(t₁ + t₂), find the relationship between the slopes of OR, PQ and the angles.</p>
Step-by-Step Solution
Key Concept: Right angle conditions at R impose constraints on the slopes that relate the angles formed by the focal chord.
<p><strong>Step 1:</strong> The slope of PQ = 2/(t₁ + t₂)</p><p><strong>Step 2:</strong> The slope of OR = -(t₁ + t₂)/2 = tan φ</p><p><strong>Step 3:</strong> Since 1/t₁ = tan θ₁ and 1/t₂ = tan θ₂:</p><p>$$\cot θ_1 + \cot θ_2 = -2\tan φ$$</p><p>∴ Answer is (a).</p>
Correct Answer: A