Points lie on a circle with a centre . Find the values of y. Hence find the radius of the circle.

All radii are of the same length in a circle.

∴ OA = OB

_{Squaring on both sides, we get}

_OA² = OB²

We know that the distance between P(x₁, y₁ ) and Q(x₂, y₂ ) is .

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

⇒ 3² + (-4y)² = 3² + (-3y – 7)²

⇒ 16y² = 9y² + 49 + 42y

⇒ 16y² - 9y² - 49 - 42y = 0

⇒ 7y² – 42y – 49 = 0

Dividing by 7,

⇒ y² – 6y – 7 = 0

By factorization method,

⇒ y² + y – 7y – 7 = 0

⇒ y(y + 1) – 7(y + 1) = 0

⇒ (y + 1) (y – 7) = 0

⇒ (y + 1) = 0 (or) (y – 7) = 0

⇒ y = -1 (or) y = 7

Case 1: When y = 7

Radius,

⇒

⇒

⇒

⇒ √793

OA = 28.16 units

Case 2: When y = -1

Radius,

⇒

⇒

⇒ √25

OA = 5 units

The values of y are 7 and -1 and the radius of circle is √793 (or) 5 units.