Passage: Consider the ellipses given by x^2 + 4y^2 = 1 and 4x^2 + y^2 = 1 . Question: If is the area of the common region that lies inside both the given ellipses, then the value of is ___________.

Step-by-step solution

The given ellipses are E₁: x^2 + 4y^2 = 1 and E₂: 4x^2 + y^2 = 1 . By symmetry, the common region is symmetric about both the coordinate axes and the lines y = x and y = -x . The total area is 8 times the area of the region in the first quadrant bounded by = 0 and = 4 . In polar coordinates, substituting x = r and y = r into the equation of E₂ (which is the inner boundary for 0 4 ), we get: 4r^2 ^2 + r^2 ^2 = 1 r^2 = 1 4 ^2 + ^2 The area of this sector is given by: A = 1 2 ₀^ 4 r^2 d = 1 2 ₀^ 4 1 4 ^2 + ^2 d Dividi