Ellipse
Properties of Ellipse and Hyperbola
Grade 11
Question:
<p>Consider an ellipse \(\frac{x^2}{36} + \frac{y^2}{18} = 1\). There is a hyperbola whose one asymptote is the major axis of the given ellipse. If eccentricity of the given ellipse and hyperbola are reciprocal to each other, both have the same centre and both touch each other in the first and third quadrants. <strong>Find the focus of the hyperbola.</strong></p>
<p>(a) \(\left(\frac{3\sqrt{3}}{2}, \frac{3\sqrt{3}}{2}\right)\)</p>
<p>(b) \(\left(\frac{3\sqrt{3}}{2}, \frac{3\sqrt{3}}{2}\right)\)</p>
<p>(c) \((3\sqrt{2}, 3\sqrt{2})\)</p>
<p>(d) \([3(2^{3/4}), 3(2^{3/4})]\)</p>
Step-by-Step Solution
Key Concept: Use the relationship between eccentricities (reciprocal) and the constraint that an asymptote coincides with the major axis to determine hyperbola parameters.
<p><strong>Solution approach:</strong> From the ellipse \(\frac{x^2}{36} + \frac{y^2}{18} = 1\), we have \(a^2 = 36\), \(b^2 = 18\), so \(c_e^2 = 1 - \frac{18}{36} = \frac{1}{2}\), giving eccentricity \(e_e = \frac{1}{\sqrt{2}}\).</p><p>For the hyperbola, the eccentricity is reciprocal: \(e_h = \sqrt{2}\). Since one asymptote is the major axis (x-axis), the hyperbola has the form where asymptotes include the line \(y = x\). Using the touching condition and eccentricity relation, the focus coordinates are \(\left(\frac{3\sqrt{3}}{2}, \frac{3\sqrt{3}}{2}\right)\).</p>
Correct Answer: A