Ellipse
Inscribed and Circumscribing Ellipses
Grade 11
Question:
<p>The ellipse \(E_1: \frac{x^2}{9} + \frac{y^2}{4} = 1\) is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse \(E_2\) passing through the point \((0, 4)\) circumscribes the rectangle R. The eccentricity of the ellipse \(E_2\) is:</p>
<p>(a) \(\frac{\sqrt{2}}{2}\)</p>
<p>(b) \(\frac{\sqrt{3}}{2}\)</p>
<p>(c) \(\frac{1}{2}\)</p>
<p>(d) \(\frac{3}{4}\)</p>
Step-by-Step Solution
Key Concept: The rectangle inscribed in \(E_1\) has vertices at \((\pm 3, \pm 2)\). The circumscribing ellipse \(E_2\) passes through these vertices and the point \((0, 4)\); use these conditions to find the equation and eccentricity.
<p>The correct answer is (c) \(\frac{1}{2}\)</p>
Correct Answer: c