a)
x² + y² - 2x + 4y + 5 = 0
⇔ (x² - 2x + 1) + (y² + 4y + 4) = 0
⇔ (x - 1)² + (y + 2)² = 0
⇔ x - 1 = 0 ∧ y + 2 = 0
⇔ x = 1 ∧ y = -2
b)
x² + 4x + y² + 6y + 13 = 0
⇔ (x² + 4x + 4) + (y² + 6y + 9) = 0
⇔ (x + 2)² + (y + 3)² = 0
⇔ x + 2 = 0 ∧ y + 3 = 0
⇔ x = -2 ∧ y = -3
c)
4x² + 9y² - 16x - 36y + 52 = 0
⇔ (4x² - 16x + 16) + (9y² - 36y + 36) = 0
⇔ (2x - 4)² + (3y - 6)² = 0
⇔ 2x - 4 = 0 ∧ 3y - 6 = 0
⇔ 2x = 4 ∧ 3y = 6
⇔ x = 2 ∧ y = 2
d)
x² + y² + x - 3y + 5/2 = 0⏐× 4
⇔ 4x² + 4y² + 4x - 12y + 10 = 0
⇔ (4x² + 4x + 1) + (4y² - 12y + 9) = 0
⇔ (2x + 1)² + (2y - 3)² = 0
⇔ 2x + 1 = 0 ∧ 2y - 3 = 0
⇔ 2x = -1 ∧ 2y = 3
⇔ x = -1/2 ∧ y = 3/2
