Hyperbola
Chord of Hyperbola
Grade 11
Question:
<p>The equation of the chord joining two points \((x_1, y_1)\) and \((x_2, y_2)\) on the rectangular hyperbola \(xy = c^2\) is</p>
<p>\(\dfrac{x}{x_1 + x_2} + \dfrac{y}{y_1 + y_2} = 1\)</p>
<p>\(\dfrac{x}{x_1 - x_2} + \dfrac{y}{y_1 - y_2} = 1\)</p>
<p>\(\dfrac{x}{y_1 + y_2} + \dfrac{y}{x_1 + x_2} = 1\)</p>
<p>\(\dfrac{x}{y_1 - y_2} + \dfrac{y}{x_1 - x_2} = 1\)</p>
Step-by-Step Solution
Key Concept: For a rectangular hyperbola xy = c², the chord joining two points can be derived using the parametric form where any point is (ct, c/t), making the chord equation linear in the parameters t₁ and t₂.
<p><strong>Step 1:</strong> Use the parametric representation of rectangular hyperbola xy = c². Any point on it can be written as (ct, c/t).</p><p><strong>Step 2:</strong> Let the two points be P(ct₁, c/t₁) and Q(ct₂, c/t₂).</p><p><strong>Step 3:</strong> The equation of line through these points: (y - c/t₁)/(x - ct₁) = (c/t₂ - c/t₁)/(ct₂ - ct₁)</p><p><strong>Step 4:</strong> Simplify the RHS: = c(1/t₂ - 1/t₁)/(c(t₂ - t₁)) = (t₁ - t₂)/(t₁t₂(t₂ - t₁)) = -1/(t₁t₂)</p><p><strong>Step 5:</strong> Cross multiply: y·t₁ - c = -x/(t₁t₂) + ct₁/(t₁t₂)</p><p><strong>Step 6:</strong> Rearranging gives: <strong>x(t₁ + t₂) - y·t₁t₂ = c(t₁ + t₂)</strong></p><p><strong>Alternative form:</strong> x(t₁ + t₂) + c²(1/y)(t₁ + t₂) = c²(t₁ + t₂), or more commonly: <strong>(t₁ + t₂)(x + c²/y) = c²(t₁ + t₂)</strong></p><p>∴ Answer: A</p>
Correct Answer: A