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

Explanation

Here, `" "OA=OB" "`(radii of a circle)
`rArr" "OA^(2)=OB^(2)`
`rArr" "(2+1)^(2)+(-3y-y)^(2)=(2-5)^(2)+(-3y-7)^(2)`
`rArr" "9+16y^(2)=9+9y^(2)+42y+49`

`rArr" "7y^(2)-42y-49=0`
`rArr " "y^(2)-6y-7=0`
`rArr" "y^(2)-7y+y-7=0`
`rArr" "y(y-7)+1(y-7)=0`
`rArr" "y-7=0" "or" "y+1=0`
`rArr" "y=7 " "or " "y=-1`
Now, the co-ordinates of centre O are either (2, -21) or (2, 3) when centre is O(2, -21), then
`" ""radius"=OB`
`" "=sqrt((2-5)^(2)+(-21-7)^(2))`
`" "=sqrt(9+784)=sqrt(793)` units
When centre is O(2, 3), then
`" ""radius"=OB`
`" "=sqrt((2-5)^(2)+(3-7)^(2))`
`" "=sqrt(9+16)=sqrt(25)=5` units