how do you find the center of a circle with the coordinates (2,6), (-1,-3), (0,8)

Question

With out using a graph.

Answer 1

Let the circle is passing through the points P(2, 6), Q(- 1, - 3), and R(0, 8).

Let the center and radius of the circle be C(h, k) and r.

|PC| = |QC| = |RC|         (Since, the distance from the center and given three points are equal(radius))

|PC|² = |QC|² = |RC|²

(h - 2)² + (k - 6)² = (h + 1)² + (k + 3)² = (h - 0)² + (k - 8)²

(h - 2)² + (k - 6)² = (h + 1)² + (k + 3)²

h² + 4 - 4h + k² + 36 - 12k = h² + 1 + 2h + k² + 9 + 6k

6h + 18k = 30

h + 3k = 5 ----------> (1)

h² + 4 - 4h + k² + 36 - 12k = h² + k² + 64 - 16k

- 4h + 4k = 64 - 40

h - k = - 6 ----------> (2)

Solve equation (1) and (2).

h + 3k = 5

h - k = - 6

(-)_____________

4k = 11

→ k = 11/4 = 2.75.

Substitute k = 2.75 in equation (2).

h - 2.75 = - 6

→ h = - 6 + 2.75 = - 3.25.

Therefore, the center of the circle C(h, k) passing through the given points is (- 3.25, 2.75).