Answer:
1. x² + y² + 8x - 14y - 35 = 0
2. x² + y² - 2x - 8y - 47 = 0
3. x² + y² - 4x - 2y - 116 = 0
4. x² + y² + 2x + 4y - 20 = 0
5. x² + y² - 4x - 8y - 52 = 0
Step-by-step explanation:
General equation of a circle is: x² + y² + 2gx + 2fy + c = 0
with center (-g, -f) and radius √(g² + f² - c)
1. From inspection: center (-4, 7) and radius = √100 = 10
Therefore, g = 4 and f = -7
√(4² + (-7)² - c) = 10 ⇒ 65 - c = 100 ⇒ c = -35
So, x² + y² + 8x - 14y - 35 = 0
2. From inspection: center (1, 4) and radius = √64 = 8
Therefore, g = -1 and f = -4
√((-1)² + (-4)² - c) = 8 ⇒ 17 - c = 64 ⇒ c = -47
So, x² + y² - 2x - 8y - 47 = 0
3. From inspection: center (2, 1) and radius = √(11²) = 11
Therefore, g = -2 and f = -1
√((-2)² + (-1)² - c) = 11 ⇒ 5 - c = 121 ⇒ c = -116
So, x² + y² - 4x - 2y - 116 = 0
4. From inspection: center (-1, -2) and radius = √25 = 5
Therefore, g = 1 and f = 2
√(1² + 2² - c) = 5 ⇒ 5 - c = 25 ⇒ c = -20
So, x² + y² + 2x + 4y - 20 = 0
5. From inspection: center (2, 4) and radius = √72
Therefore, g = -2 and f = -4
√((-2)² + (-4)² - c) = √72 ⇒ 20 - c = 72 ⇒ c = -52
So, x² + y² - 4x - 8y - 52 = 0
