A circle has a center that falls on the line y = 5/8x +6 y=58x+6 and passes through ( 1 ,5 )(1,5) and (2 ,4 )(2,4). What is the equation of the circle?

1 Answer
Douglas K.
Oct 3, 2016

2384/361 = (x - 24/19)² + (y - 148/19)²

Explanation:

The equation of a circle is:

r² = (x - h)² + (y - k)²

where r is the radius and (h, k) is the center.

Using the given points we can use the above to create two equation:

r² = (1 - h)² + (5 - k)²
r² = (2 - h)² + (4 - k)²

Because r² = r² we can set the right sides equal:

(1 - h)² + (5 - k)² = (2 - h)² + (4 - k)²

Use the pattern (a - b)² = a² - 2ab + b² to expand the squares:

1 - 2h + h² + 25 - 10k + k² = 4 - 8h + h² + 16 - 8k + k²

Combine like terms:

6h + 6 = 2k

k = 3h + 3

Substitute the point (h, k) into the given linear equation:

k = 5/8h + 6

Because k = k we can set the right sides equal:

3h + 3 = 5/8h + 6

24h + 24 = 5h + 48

h = 24/19

k = 5/8(24/19) + 6

k = 120/152 + 1064/152

k = 120/152 + 1064/152

k = 1184/152

k = 148/19

r² = (1 - 24/19)² + (5 - 148/19)²

r² = (-5/19)² + (-53/19)²

r² = 2384/361

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