UPSC Prelims 1999·GS1·science-and-technology·science and technology

The area of an ellipse is twice that of a circle. The major diameter of the ellipse is twice that of the minor diameter. The radius of the circle is

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  1. A50% of minor diameter of the ellipseCorrect
  2. B50% of major diameter of the ellipse
  3. Cminor diameter of the ellipse
  4. Dmajor diameter of the ellipse

Explanation

Let the minor diameter of the ellipse be 2b and the major diameter be 2a. The problem states the major diameter is twice the minor diameter, so 2a equals 2 times 2b, which means a equals 2b. The area of an ellipse is pi times a times b. Substituting a with 2b, the area becomes pi times 2b times b, which is 2 times pi times b squared. The area of a circle is pi times r squared. The problem states the area of the ellipse is twice the area of the circle. Therefore, 2 times pi times b squared equals 2 times pi times r squared. By canceling out 2 and pi from both sides, we get b squared equals r squared, meaning r equals b. Since b is half of the minor diameter 2b, the radius r is equal to 50 percent of the minor diameter of the ellipse. Therefore, option A is correct.
science-and-technology: The area of an ellipse is twice that of a circle. The major diameter of the ellipse is twice that of the minor diameter.
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