Question

Points A(-1, y) and B(5, 7) lie on a circle with centre O(2, -3y) such that AB is a diameter of the circle. Find the value of y. Also, find the radius of the circle.

Answer 1

Points A(-1, y) and B(5, 7) lie on a circle with centre O(2, -3y).

Which means: OA = OB or OA² = OB²

using distance formula, we get

(-1 - 2)² + (y - (-3y))² = (5 - 2)² + (7 - (-3y))²

9 + 16y² = 9 + (7 + 3y)²

16y² = 49 + 42y + 9y²

7y² - 42y - 49 = 0

7(y² - 6y - 7) = 0

y² - 7y + y - 7 = 0

y(y-7) + 1(y - 7) = 0

(y + 1) (y - 7) = 0

Therefore, y = 7 or y = -1

When y = -1

The coordinate of O, A and B are O(2, 3), A(-1, -1) and B(5, 7) respectively.

\(\therefore\) Radius = \(OA = \sqrt{(2 + 1)^2 + (3 + 1)^2}\)

\(= \sqrt{(3)^2 + (4)^2}\)

\(= \sqrt {9 + 16}\)

\(= \sqrt{25}\)

\(= 5\)

When y = 7

The coordinate of O, A and B are O(2, -21), A(-1, 7) and B(5, 7) respectively.

\(\therefore\) Radius = \(OA = \sqrt{(2 + 1)^2 + (-21-7)^2}\)

\(= \sqrt{(3)^2 + (-28)^2}\)

\(= \sqrt{9 + 784}\)

\(= \sqrt{793}\)