Question

Points A(–1, y) and B(5, 7) lie on a circle with centre O(2, –3y). Find the values of y. Hence find the radius of the circle.

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

A and B are the two points that lie on the circle and O is the centre of the circle.

Therefore, OA and OB are the radii of the circle.

Using the distance formula, we have:

`OA=sqrt((-1-2)^2+(y+3y)^2)=sqrt(9+16y^2)`

`OB=sqrt((5-2)^2+(7+3y)^2)=sqrt(9+(7+3y)^2)`

Now, OB = OA              (Radii of the same circle)

`sqrt(9+(7+3y)^2)=sqrt(9+16y^2)`

9 + (7 + 3y)² = 9 + 16y² (squaring both the sides)

49 + 9y² + 42y = 16y²

⇒ 7y² – 42y – 49 = 0

⇒ y² – 6y – 7 = 0

⇒ y² − 7y + y – 7 = 0

⇒ (y − 7)(y + 1) = 0

⇒ y – 7 = 0 or y + 1 = 0

⇒ y = 7 or y = –1

When y = 7:

Radius of the circle, `OA=sqrt(9+16y^2)=sqrt(9+16×49)=sqrt(793)`

When y = –1:

Radius of the circle, `OA=sqrt(9+16y^2) =sqrt(9+16×1)=sqrt(25)=5`