<p>If the line joining foci subtends an angle of \(90°\) at an extremity of minor axis then the eccentricity of the ellipse is:</p>
<p>(a) \(\frac{1}{3}\)</p>
<p>(b) \(\frac{1}{2}\)</p>
<p>(c) \(\frac{1}{\sqrt{2}}\)</p>
<p>(d) none of these</p>
Step-by-Step Solution
Key Concept: When a focal chord subtends a right angle at a point on the ellipse, the dot product of the vectors from that point to the foci equals zero.
<p>Let the extremity of minor axis be \(B(0, b)\). The foci are at \(F_1(-c, 0)\) and \(F_2(c, 0)\) where \(c = ae\).</p><p>For the angle \(\angle F_1BF_2 = 90°\):</p><p>\(\vec{BF_1} \cdot \vec{BF_2} = 0\)</p><p>\((-c, -b) \cdot (c, -b) = 0\)</p><p>\(-c^2 + b^2 = 0\)</p><p>\(b^2 = c^2\)</p><p>\(b^2 = a^2e^2\)</p><p>Since \(b^2 = a^2(1-e^2)\):</p><p>\(a^2(1-e^2) = a^2e^2\)</p><p>\(1 - e^2 = e^2\)</p><p>\(e^2 = \frac{1}{2}\)</p><p>\(e = \frac{1}{\sqrt{2}}\)</p><p>∴ Answer is (c).</p>
Correct Answer: b