Let the new origin be (h, k) = (1, -2)

Then, the transformation formula become:

x = X + 1 and y = Y + (-2) = Y – 2

Substituting the value of x and y in the given equation, we get

2x² + y² – 4x + 4y = 0

Thus,

2(X + 1)² + (Y – 2)² – 4(X + 1) + 4(Y – 2) = 0

⇒ 2(X² + 1 + 2X) + (Y² + 4 – 4Y) – 4X – 4 + 4Y – 8 = 0

⇒ 2X² + 2 + 4X + Y² + 4 – 4Y – 4X + 4Y – 12 = 0

⇒ 2X² + Y² – 6 = 0

⇒ 2X² + Y² = 6

Hence, the transformed equation is 2X² + Y² = 6