Find the centre of the circle passing through (2, 1), (5, - 8) and (2,

1.	Find the centre of the circle passing through (2, 1), (5, - 8) and (2, - 9).
Answer» Coordinates of points on a circle are A(2,1), B(5,-8) and C(2,-9). Let the coordinates of the centre of the circle be O(x, y) Using distance formula = \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) Since the distance of the points A, B and C will be equal from the center, therefore ⇒ OA = OB \(\sqrt{(x - 2)^2 + (y - 1)^2}\) = \(\sqrt{(x - 5)^2 + (y + 8)^2}\) On squaring both sides, we get ⇒ x² + 4 - 4x + y² + 1 - 2y = x² + 25 - 10x + y²+ 64 + 16y ⇒ 6x - 18y - 84 = 0 ⇒ x - 3y - 14 = 0 ------------- (1) Similarly, OC = OB \(\sqrt{(x - 2)^2 + (y + 9)^2}\) = \(\sqrt{(x - 5)^2 + (y + 8)^2}\) ⇒ x² + 4 - 4x + y² + 81 + 18y = x² + 25 - 10x + y² + 64 + 16y ⇒ 6x - 2y - 4 = 0 ⇒ 3x - y - 2 = 0 ------------- (2) By solving equations (1) and (2), we get x = -1, y = -5 So, the coordinates of the centre of the circle is (-1, -5). Radius of the circle = OA = \(\sqrt{(-1 - 2)^2 + (-5 - 1)^2}\) = \(\sqrt{9+36}\) = \(\sqrt{45}\) = \(3\sqrt5\) units

Find the centre of the circle passing through (2, 1), (5, - 8) and (2, - 9).

Answer»

Coordinates of points on a circle are A(2,1), B(5,-8) and C(2,-9).

Let the coordinates of the centre of the circle be O(x, y)

Using distance formula = \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

Since the distance of the points A, B and C will be equal from the center, therefore

⇒ OA = OB

\(\sqrt{(x - 2)^2 + (y - 1)^2}\) = \(\sqrt{(x - 5)^2 + (y + 8)^2}\)

On squaring both sides, we get

⇒ x² + 4 - 4x + y² + 1 - 2y = x² + 25 - 10x + y²+ 64 + 16y

⇒ 6x - 18y - 84 = 0

⇒ x - 3y - 14 = 0 ------------- (1)

Similarly, OC = OB

\(\sqrt{(x - 2)^2 + (y + 9)^2}\) = \(\sqrt{(x - 5)^2 + (y + 8)^2}\)

⇒ x² + 4 - 4x + y² + 81 + 18y

= x² + 25 - 10x + y² + 64 + 16y

⇒ 6x - 2y - 4 = 0

⇒ 3x - y - 2 = 0 ------------- (2)

By solving equations (1) and (2), we get x = -1, y = -5

So,

the coordinates of the centre of the circle is (-1, -5).

Radius of the circle = OA = \(\sqrt{(-1 - 2)^2 + (-5 - 1)^2}\)

= \(\sqrt{9+36}\)

= \(\sqrt{45}\)

= \(3\sqrt5\) units