<p>The ellipse \(\frac{x^2}{2} + \frac{y^2}{1} = 1\) has foci \(S_1\) and \(S_2\). Rectangle \(S_1PS_2Q\) is completed (where \(P\) and \(Q\) are on the ellipse). Which of the following is/are correct?</p>
<p>(a) Number of such pair \(P, Q\) is one</p>
<p>(b) Area of rectangle \(S_1PS_2Q\) is equal to 2 sq. units</p>
<p>(c) There will be infinite such pairs \(P, Q\)</p>
<p>(d) Rectangle \(S_1PS_2Q\) is a square</p>
Step-by-Step Solution
Key Concept: Use properties of foci and symmetry of ellipse to construct rectangles with vertices on the ellipse. Recognize that infinitely many such rectangles exist.
<p>For the ellipse \(\frac{x^2}{2} + \frac{y^2}{1} = 1\), we have \(a^2 = 2\), \(b^2 = 1\), so \(c^2 = a^2 - b^2 = 1\), giving \(c = 1\). The foci are \(S_1(-1, 0)\) and \(S_2(1, 0)\).</p><p>For rectangle \(S_1PS_2Q\) with \(P\) and \(Q\) on the ellipse, if \(P = (x, y)\), then \(Q = (x, -y)\) by symmetry. The length \(S_1S_2 = 2\) and width = \(2y\). There are infinite such pairs as \(y\) varies, giving area = \(2 \times 2y = 4y\). For the specific configuration where the rectangle has maximum area, this gives 2 sq. units.</p>
Correct Answer: b, c