Question

Aflati numerele reale x si y daca a)x²+y²-2x+4y+5=0
b)x²+4x+y²+6y+13=0
c)4x²+9y²-16x-36y+52=0
d)x²+y²+x-3y+5/2=0

Answer 1

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