<p>The triangle PQR of area <i>A</i> is inscribed in the parabola <i>y</i><sup>2</sup> = 4<i>ax</i> such that the vertex P lies at the vertex of the parabola and the base QR is a focal chord. The modulus of the difference of the ordinates of the points Q and R is:</p>
<p>(a) <i>A</i>/(2<i>a</i>)</p>
<p>(b) <i>A</i>/<i>a</i></p>
<p>(c) 2<i>A</i>/<i>a</i></p>
<p>(d) 4<i>A</i>/<i>a</i></p>
Step-by-Step Solution
Key Concept: For a focal chord of parabola y² = 4ax, if points Q and R have parameters t₁ and t₂, then t₁t₂ = -1. The area of triangle with vertex at origin and base as focal chord relates to the difference in ordinates through the relationship: Area = (1/2) × base × height.
<p><strong>Step 1: Set up parametric form.</strong> For parabola y² = 4ax, any point can be written as (at², 2at). Let Q = (at₁², 2at₁) and R = (at₂², 2at₂) be points on the focal chord. Vertex P is at origin (0, 0).</p><p><strong>Step 2: Apply focal chord condition.</strong> For a focal chord of y² = 4ax, the relationship between parameters is: t₁t₂ = -1, which means t₂ = -1/t₁.</p><p><strong>Step 3: Calculate the area of triangle PQR.</strong> With P at origin, Q = (at₁², 2at₁), R = (at₂², 2at₂):<br/>Area = (1/2)|x₁y₂ - x₂y₁| = (1/2)|at₁² · 2at₂ - at₂² · 2at₁|<br/>= (1/2)|2a²t₁t₂(t₁ - t₂)|<br/>= (1/2)|2a² · (-1) · (t₁ - t₂)|<br/>= a²|t₁ - t₂|</p><p><strong>Step 4: Find |y₁ - y₂|.</strong> The ordinates are y₁ = 2at₁ and y₂ = 2at₂.<br/>|y₁ - y₂| = |2at₁ - 2at₂| = 2a|t₁ - t₂|</p><p><strong>Step 5: Relate area to |y₁ - y₂|.</strong> From Step 3: A = a²|t₁ - t₂|<br/>Therefore: |t₁ - t₂| = A/a²<br/>Substituting into Step 4:<br/>|y₁ - y₂| = 2a · (A/a²) = 2A/a</p><p><strong>∴ Answer:</strong> C</p>
Correct Answer: C