Circle Center From Three Points
The following three points lie on a circle in the x y plane. Coordinates are approximated to three decimal places.
| ( x 1 , y 1 ) | ( 8 . 5 9 0 , 9 . 2 1 3 ) |
| ( x 2 , y 2 ) | ( 3 . 0 0 0 , 1 2 . 0 0 0 ) |
| ( x 3 , y 3 ) | ( − 3 . 0 6 2 , 1 . 5 0 0 ) |
If the center of the circle has coordinates ( h , k ) , determine ( h + k ) .
Details and Assumptions:
Round
h
and
k
individually to the nearest integer before adding them.
The answer is 8.
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2 solutions
Good and easy solution too ( +1). And nice problem...
Given three points not aligned there exists one only circumference passing through of them. In this case, the circumference will have an equation C ≡ ( x − h ) 2 + ( y − k ) 2 = r 2 ≡ x 2 + y 2 − 2 h x − 2 k y + c = 0 Then, it's sufficient to substitute ( x 1 , y 1 ) , ( x 2 , y 2 ) , ( x 3 , y 3 ) in C and solve the equation for h , k , r or h , k , c ... In this case,it's better h , k , c , it's a long process, but not very hard... It's a linear system of three equations with three unknown variables. h ≈ 2 . 9 9 9 . . . , k ≈ 5 . 0 0 0 . .
I like how you kept it linear by representing the sum of the squares of h and k as a constant. I have also posted a geometric solution.