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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Q: 149 (IAS/1999)
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

question_subject: 

Maths

question_exam: 

IAS

stats: 

0,3,6,3,2,4,0

keywords: 

{'ellipse': [0, 1, 0, 2], 'major diameter': [0, 1, 0, 0], 'minor diameter': [0, 1, 0, 0], 'radius': [0, 0, 2, 2], 'circle': [0, 0, 2, 1], 'area': [0, 0, 0, 1]}

Let`s assume the major diameter of the ellipse is denoted by D and the minor diameter is denoted by d.

The formula for the area of an ellipse is A = ? * (D/2) * (d/2), and the formula for the area of a circle is A = ? * r^2, where r is the radius of the circle.

Given that the area of the ellipse is twice that of the circle, we can write the equation:

? * (D/2) * (d/2) = 2 * ? * r^2

Canceling out ? and simplifying, we have:

(D/2) * (d/2) = 2 * r^2

Dividing both sides by 2, we get:

(D/4) * (d/4) = r^2

Taking the square root of both sides, we have:

r = sqrt((D/4) * (d/4))

Since the major diameter D is twice the minor diameter d, we can substitute D = 2d into the equation:

r = sqrt((2d/4) * (d/4))

= sqrt((d/2) * (d/4))

= sqrt(d^2 / 8)

= d / sqrt(8)

= d / (2 * sqrt(2))

Therefore, the radius of the circle is d / (2 * sqrt(2)), which is equal to approximately 0.3536 times the minor diameter of the ellipse.