<p>Rectangle \(ABCD\) has area \(200\). An ellipse with area \(200\pi\) passes through \(A\) and \(C\) and has foci at \(B\) and \(D\). Let perimeter of the rectangle \(ABCD\) is \(P\), then \(\frac{P}{10} =\)</p>
Step-by-Step Solution
Key Concept: For an ellipse with foci at opposite vertices of a rectangle, use the property that any point on the ellipse satisfies the sum of distances to the foci equals 2a. Combined with the area formula πab = 200π, we can determine the rectangle's dimensions.
<p><strong>Step 1: Set up coordinates.</strong> Let rectangle ABCD have dimensions length = 2p and width = 2q, with center at origin. Place vertices: A(p, q), B(-p, q), C(-p, -q), D(p, -q).</p><p><strong>Step 2: Identify foci locations.</strong> Foci are at B(-p, q) and D(p, -q). The distance between foci is: BD = √[(2p)² + (2q)²] = 2√(p² + q²). Therefore, 2c = 2√(p² + q²), giving c = √(p² + q²).</p><p><strong>Step 3: Use the focal property.</strong> Point A(p, q) lies on the ellipse. Sum of distances from A to foci B and D: AB + AD = 2a (definition of ellipse). Calculate: AB = √[(2p)² + 0²] = 2p and AD = √[0² + (2q)²] = 2q. Therefore: 2a = 2p + 2q, so a = p + q.</p><p><strong>Step 4: Find relationship between a, b, c.</strong> For an ellipse: c² = a² - b². We have c² = p² + q² and a² = (p + q)². Therefore: p² + q² = (p + q)² - b², which gives b² = p² + 2pq + q² - p² - q² = 2pq, so b = √(2pq).</p><p><strong>Step 5: Use the area constraint.</strong> Area of ellipse: πab = 200π. Therefore: ab = 200. Substituting: (p + q)·√(2pq) = 200.</p><p><strong>Step 6: Use the rectangle area constraint.</strong> Area of rectangle: (2p)(2q) = 200, so pq = 50.</p><p><strong>Step 7: Solve for p + q.</strong> From Step 5: (p + q)·√(2·50) = 200, which gives (p + q)·√100 = 200, so (p + q)·10 = 200, therefore p + q = 20.</p><p><strong>Step 8: Calculate rectangle perimeter.</strong> Perimeter P = 2(2p + 2q) = 4(p + q) = 4(20) = 80.</p><p><strong>Step 9: Find P/10.</strong> P/10 = 80/10 = 8.</p><p><strong>∴ Answer: 8</strong></p>
Correct Answer: 8