Hyperbola
Directrices of Hyperbola
Grade 11
Question:
<p>Consider a hyperbola H whose centre is at the origin and the line \(x + y = 2\) touches it at point \((1, 1)\). The tangent \(x + y = 2\) intersects the asymptotes of H at points A and B such that the length of segment \(AB = 6\sqrt{2}\). <strong>Find the equation of the pair of directrices of H.</strong></p>
<p>(a) \(x^2 + y^2 + 2xy - 1 = 0\)</p>
<p>(b) \(5x^2 + 5y^2 + 10xy - 4 = 0\)</p>
<p>(c) \(5x^2 + 5y^2 + 10xy - 2 = 0\)</p>
<p>(d) \(5x^2 + 5y^2 + 10xy - 6 = 0\)</p>
Step-by-Step Solution
Key Concept: Use the tangency point, tangent line equation, and the geometry of asymptotes intersecting the tangent to determine the hyperbola parameters and hence the directrices.
<p><strong>Solution approach:</strong> Since the hyperbola is centred at the origin with tangent \(x + y = 2\) at \((1,1)\), and the asymptotes intersect this tangent with segment length \(6\sqrt{2}\), we can determine the hyperbola's parameters. The asymptotes pass through the origin. Using the constraint that the distance from origin to the tangent line is \(\frac{2}{\sqrt{2}} = \sqrt{2}\), combined with the asymptote-tangent intersection distance condition, the hyperbola equation can be determined. The directrices form a pair given by the equation \(5x^2 + 5y^2 + 10xy - 2 = 0\).</p>
Correct Answer: C