Answer:
(x − 4)² + (y − 3)² = 25
Step-by-step explanation:
The equation of a circle is:
(x − h)² + (y − k)² = r²
Given three points on the circle, we can write three equations:
(1 − h)² + (7 − k)² = r²
(8 − h)² + (6 − k)² = r²
(7 − h)² + (-1 − k)² = r²
Expanding:
1 − 2h + h² + 49 − 14k + k² = r²
64 − 16h + h² + 36 − 12k + k² = r²
49 − 14h + h² + 1 + 2k + k² = r²
Simplifying:
50 − 2h + h² − 14k + k² = r²
100 − 16h + h² − 12k + k² = r²
50 − 14h + h² + 2k + k² = r²
Subtracting the first equation from the second and third equations:
50 − 14h + 2k = 0
-12h + 16k = 0
Solving the system of equations, first reduce:
25 − 7h + k = 0
-3h + 4k = 0
Solve with substitution or elimination. Using substitution, solve for k in the first equation and substitute into the second.
k = 7h − 25
-3h + 4(7h − 25) = 0
-3h + 28h − 100 = 0
25h = 100
h = 4
k = 7h −25
k = 7(4) − 25
k = 3
Now plug these into any of the original three equations to find r.
(1 − h)² + (7 − k)² = r²
(1 − 4)² + (7 − 3)² = r²
9 + 16 = r²
25 = r²
The equation of the circle is:
(x − 4)² + (y − 3)² = 25
Graph: desmos.com/calculator/ctoljeqhnp
