Question

O centro da circunferência x2 + y2 + 5x + 4y – 25 = 0 é:

Answer 1

Podemos resolver de duas formas

1ª Forma : Completando quadrados

$\text x^2 + \text y^2 + 5\text x + 4\text y - 25 = 0$

completando os quadrados:

$\displaystyle \text x^2 + 5\text x + \frac{25}{4}+\text y^2 + 4\text y + 4 = 25 + 4 + \frac{25}{4}$  

$\displaystyle (\text x +\frac{5}{2})^2+(\text y+2)^2 = \frac{100+16+25}{4}$

$\displaystyle (\text x +\frac{5}{2})^2+(\text y+2)^2 = \frac{141}{4}$

Portanto :

$\huge\boxed{\text{Centro :} \ (\ \frac{-5}{2} , -2\ ) }$

2ª Forma : Derivando parcialmente para achar o centro

$\displaystyle \frac{\partial }{\partial \text x}(\text x^2+\text y^2 + 5\text x + 4 \text y - 25=0 ) \to 2\text x + 5 = 0$

$\displaystyle \frac{\partial }{\partial \text y}(\text x^2+\text y^2 + 5\text x + 4 \text y - 25=0 ) \to 2\text y + 4 = 0$

Portanto :

$2\text x + 5 = 0 \to \displaystyle \text x = \frac{-5}{2}$

$2\text y + 4 = 0 \to \text y = -2$  

$\huge\boxed{\text{Centro :} \ (\ \frac{-5}{2} , -2\ ) }$