Ellipse
Eccentricity
Grade 11
Question:
<p>The eccentricity of the ellipse which meets the straight line \(\frac{x}{7} + \frac{y}{2} = 1\) on the axis of x and the straight line \(\frac{x}{3} - \frac{y}{5} = 1\) on the axis of y and whose axis lie along the axes of coordinates is:</p>
<p>(a) \(\frac{3\sqrt{2}}{7}\)</p>
<p>(b) \(\frac{2\sqrt{6}}{7}\)</p>
<p>(c) \(\frac{2\sqrt{3}}{7}\)</p>
<p>(d) none of these</p>
Step-by-Step Solution
Key Concept: Find where the given lines intersect the coordinate axes to determine the semi-major and semi-minor axes of the ellipse.
<p>The line \(\frac{x}{7} + \frac{y}{2} = 1\) meets the x-axis at \((7, 0)\), so \(a = 7\).</p><p>The line \(\frac{x}{3} - \frac{y}{5} = 1\) meets the y-axis when \(x = 0\): \(-\frac{y}{5} = 1 \Rightarrow y = -5\), so \(b = 5\).</p><p>The ellipse is \(\frac{x^2}{49} + \frac{y^2}{25} = 1\)</p><p>Eccentricity: \(e = \sqrt{1 - \frac{b^2}{a^2}} = \sqrt{1 - \frac{25}{49}} = \sqrt{\frac{24}{49}} = \frac{2\sqrt{6}}{7}\)</p><p>∴ Answer is (b).</p>
Correct Answer: b