The above two curves touch each other at and where .
Let the area enclosed between the two curve be and for the case and respecively. Find the value of .
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The symmetric matrix of the quadratic form q ( x , y ) = x 2 − 2 x y + 3 y 2 is A = [ 1 − 1 − 1 3 ] and the area of the ellipse q ( x , y ) = k is det A k π = 2 k π . Using the values k = 2 ± 2 from the first part , we find M = 2 π , m = ( 2 − 2 ) π and m M = 1 + 2 ≈ 2 . 4 1 4 . The answer we seek is 2 4 1 4