Ellipse
Confocal Ellipse and Hyperbola
Grade 11
Question:
<p>If the ellipse \(x^2 + k^2y^2 = k^2a^2\) is confocal with the hyperbola \(x^2 - y^2 = a^2\) (\(k > 1\)), then which of the following statement(s) is/are correct?</p>
<p>(a) Ratio of eccentricities of ellipse and hyperbola is \(\frac{1}{3}\)</p>
<p>(b) Ratio of major axis of ellipse and transverse axis of hyperbola is \(\sqrt{3}\)</p>
<p>(c) Ratio of minor axis of ellipse and conjugate axis of hyperbola is \(\sqrt{3}\)</p>
<p>(d) Ratio of length of latus rectum of ellipse and hyperbola is \(\frac{1}{3}\)</p>
Step-by-Step Solution
Key Concept: Use the confocal condition (equal foci distance) to find the relationship between parameters, then compute the required ratios.
<p><strong>Step 1:</strong> For the ellipse \(x^2 + k^2y^2 = k^2a^2\), rewrite as \(\frac{x^2}{k^2a^2} + \frac{y^2}{a^2} = 1\). Since \(k > 1\), major axis is along \(x\), with \(A^2 = k^2a^2\), \(B^2 = a^2\).</p><p><strong>Step 2:</strong> Eccentricity of ellipse: \(e_e^2 = 1 - \frac{a^2}{k^2a^2} = 1 - \frac{1}{k^2}\), so \(c_e^2 = k^2a^2 - a^2 = a^2(k^2 - 1)\).</p><p><strong>Step 3:</strong> For hyperbola \(x^2 - y^2 = a^2\): \(c_h^2 = a^2 + a^2 = 2a^2\), so \(c_h = a\sqrt{2}\).</p><p><strong>Step 4:</strong> Confocal condition: \(a^2(k^2 - 1) = 2a^2\) gives \(k^2 = 3\), so \(k = \sqrt{3}\).</p><p><strong>Step 5:</strong> Calculate ratios using \(k = \sqrt{3}\).</p><p>∴ Answer is (a, b, d).</p>
Correct Answer: a, b, d