What are the coordinates of the center of the circle that passes through the points (1, 1), (1, 5), and (5, 5)?

Solution:

Let the equation of circle is x² + y² + 2gx + 2fy + c = 0 --- (1)

Step-1:

If the point passes through (1, 1)

1² + 1² + 2g(1) + 2f(1) + c = 0

1 + 1 + 2g + 2f + c = 0

2g + 2f + c + 2 = 0 --- (2)

Step-2:

If the point passes through (1, 5)

1² + 5² + 2g(1) + 2f(5) + c = 0

1 + 25 + 2g + 10f + c = 0

2g + 10f + c + 26 = 0 --- (3)

Step-3:

If the point passes through (5, 5)

5² + 5² + 2g(5) + 2f(5) + c = 0

25 + 25 +10g +10f + c = 0

10g + 10f + c + 50 = 0 --- (4)

Step- 4:

Subtract (1) - (2)

⇒ (2 - 2)g + (2 -10)f + (c - c) + (2 - 26) = 0 --- (5)

⇒ -8f - 24 = 0

⇒f = 24/-8 = -3

Step-5:

Subtract (3) - (4)

⇒ (2 - 10)g + (10 - 10)f + 26 - 50 = 0

⇒ -8g - 24 = 0

⇒ g = 24/(-8) = -3

Step-6:

(4) with g = -3 and f = -3

⇒ 10(-3) + 10(-3) + c + 50 = 0

⇒ -30 - 30 + c + 50 = 0

⇒ c - 10 = 0

⇒ c = 10

Therefore, the equation of circle is x² + y² - 6x - 6f + 10 = 0

The center of the circle is (-g, -f)

Therefore, the center of the circle is (3, 3).

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What are the coordinates of the center of the circle that passes through the points (1, 1), (1, 5), and (5, 5)?

Summary:

The coordinates of the center of the circle that passes through the points (1, 1), (1, 5), and (5, 5) is (3,3).