A circle has a center that falls on the line y = 3/7x +1 y=37x+1 and passes through ( 2 ,8 )(2,8) and (3 ,5 )(3,5). What is the equation of the circle?

1 Answer
Douglas K.
Oct 3, 2016

The equation is:

2405 = (x - 49)² + (x - 22)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 point.

Using the two given points, we write two equations:

r² = (2 - h)² + (8 - k)²
r² = (3 - h)² + (5 - k)²

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

(2 - h)² + (8 - k)²= (3 - h)² + (5 - k)²

Using the pattern, (a - b)² = a² - 2ab + b² we expand the squares:

4 - 4h + h² + 64 - 16k + k²= 9 - 6h + h² + 25 - 10k + k²

4 - 4h + 64 - 16k= 9 - 6h + 25 - 10k

4 + 64 -6k= 9 - 2h + 25

-6k = -2h - 34

k = 1/3h + 17/3

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

k = 3/7h + 1

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

3/7h + 1 = 1/3h + 17/3

9h + 21 = 7h + 119

2h = 98

h = 49

k = 3/7(49) + 1

k = 22

Substitute the center into the first equation and solve for r²:

r² = (2 - 49)² + (8 - 22)²

r² = (-47)² + (-14)²

r² = 2405

The equation is:

2405 = (x - 49)² + (x - 22)²

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