Let a = side of square A;

Let b = side of square B;

Perimeter of B = 4b

Perimeter of A = 4a

Perimeter of B = 2 * Perimeter of A

4b = 2 * 4a;

b = 2a;

From Pythagoras rule, length of a diagonal equals \sqrt{2} * side of the square:

Diagonal of square A = \(\sqrt{2} * a\)

Diagonal of square B = \(\sqrt{2} * b = \sqrt{2} * 2a = \sqrt{8} * a\)

Radius of circle X = Half of Diagonal of square A, since square A is inscribed in circle X = \(\sqrt{2} * a/2;\)

Radius of circle Y = Half of Diagonal of square B, since square B is inscribed in circle Y =\(\sqrt{8} * a/2;\)

Area of circle X = \(Pi * r^2 = Pi * (\sqrt{2} * a/2)^2 = Pi * a/2;\)

Area of circle Y = \(Pi * r^2 = Pi * (\sqrt{8} * a/2)^2 = Pi * 2a;\)

Area of circle Y = \(k *\)Area of circle X

\(Pi * 2a = k * Pi * a/2\)

\(2 = k * 1/2\)

\(k = 4;\)

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