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

#### Solution

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+16y^^{2} (squaring both the sides)

49+9y^{2}+42y=16y^{2}

⇒7y^{2}−42y−49=0

⇒y^{2}−6y−7 =0

⇒y^{2}−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`