<p>In a parabola <i>y</i><sup>2</sup> = 4<i>ax</i> the angle <i>θ</i> that the latus rectum subtends at the vertex of the parabola is:</p>
<p>(a) dependent on the length of the latus rectum</p>
<p>(b) independent of the latus rectum and lies between 5<i>π</i>/6 and <i>π</i></p>
<p>(c) independent of the latus rectum and lies between 3<i>π</i>/4 and 5<i>π</i>/6</p>
<p>(d) independent of the latus rectum and lies between 2<i>π</i>/3 and 3<i>π</i>/4</p>
Step-by-Step Solution
Key Concept: The latus rectum of y² = 4ax has endpoints at (a, 2a) and (a, -2a). We need to find the angle these endpoints subtend at the vertex (origin) using the dot product formula, then verify this angle is independent of 'a'.
<p><strong>Step 1: Identify the latus rectum endpoints</strong></p><p>For parabola y² = 4ax, the latus rectum passes through the focus F(a, 0) and is perpendicular to the axis. The endpoints are L₁ = (a, 2a) and L₂ = (a, -2a).</p><p><strong>Step 2: Find vectors from vertex to endpoints</strong></p><p>Vector OL₁ = (a, 2a) and Vector OL₂ = (a, -2a), where O is the vertex at origin (0, 0).</p><p><strong>Step 3: Apply dot product formula</strong></p><p>OL₁ · OL₂ = a² - 4a² = -3a²</p><p>|OL₁| = √(a² + 4a²) = a√5</p><p>|OL₂| = √(a² + 4a²) = a√5</p><p><strong>Step 4: Calculate the angle</strong></p><p>cos θ = (OL₁ · OL₂)/(|OL₁| × |OL₂|) = -3a²/(a√5 × a√5) = -3a²/5a² = -3/5</p><p><strong>Step 5: Verify independence from 'a'</strong></p><p>The angle depends only on cos θ = -3/5, which is independent of parameter 'a'. Therefore θ = cos⁻¹(-3/5) ≈ 126.87° ≈ 2.214 radians.</p><p><strong>Step 6: Check the range</strong></p><p>2π/3 ≈ 2.094 radians and 3π/4 ≈ 2.356 radians. Since 2.094 < 2.214 < 2.356, the angle lies between 2π/3 and 3π/4.</p><p><strong>∴ Answer: D</strong></p>
Correct Answer: D