Question

Determine a equação da circunferência que passa pelos pontos (4,5), (3,-2) e (1,-4).

Answer 1

x ² + y ² + D x + E y + F = 0

(4, 5)

4² + 5² + 4D + 5E + F = 0
4D + 5E + F =- 41

(3, -2)

3² (-2)² + 3D - 2E + F = 0

3D - 2E + F = -13

(1, -4)

1² + (-4)² + D -4E + F = 0
D - 4E + F =- 17

formando o sistema

{ 4D + 5E + F = -41
{ 3D - 2E + F = -13
{ D - 4E + F = -17

Aplicando Gauss

|4....5.....1...|-41
|3...-2.....1...|-13
|1....-4.....1...|-17

f2 = -3f1 + 4f2
f3 = f1 - 4f3

|4....5.....1.....|-41
|0...-23....1...|71
|0.....21....-3...|27

f3 = 21f2 + 23f3

|4....5.....1.....|41
|0...-23....1...|71
|0.....0....-48...|2112

-48F = 2112
F = 2112/-48
F = -44

-23E + F = 71
-23E = 71 + 44
-23E = 115
E = 115/-23
E = -5

4D + 5E + F =- 41
4D + 5•(-5) - 44 =- 41
4D = -41 +69
D = 28/4
D = 7

x ² + y ² + D x + E y + F = 0

x² + y² +7x - 5y - 44 = 0

x² + 7x + y² - 5y = 44