SOLUTION: A circle is inscribed in a quadrant of a larger circle. Find the ratio of the area of the small circle to that of the quadrant. Thanks

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Question 332966: A circle is inscribed in a quadrant of a larger circle. Find the ratio of the area of the small circle to that of the quadrant.
Thanks
Answer by Alan3354(69443) (Show Source):
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A circle is inscribed in a quadrant of a larger circle. Find the ratio of the area of the small circle to that of the quadrant.
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The larger circle has a radius R.
This circle about the Origin is x^2 + y^2 = R^2
The smaller circle has radius r. Its center will be on the line x = y in the 1st quadrant.
The smaller circle's eqn is (x-h)^2 + (y-h)^2 = r^2
R = r(1 + sqrt(2))
The area is a function of the square of the radii
--> Area of the large circle = area*(1 + sqrt(2))^2
Area of the quadrant = (area*(1 + sqrt(2))^2)/4
AQ/a = (1 + 2 + 2sqrt(2))/4 = (3 + 2sqrt(2))/4
--> ratio of small circle to quadrant = 4/(3 + 2sqrt(2))
= 4(3 - 2sqrt(2)) to 1
=~ 0.68629