# One of the diameter of circle x^2 + y^2 - 2x - 6y + 6 = 0 is a chord to the circle with centre ( 2, 1 ). Then find the radius of circle.

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You may find the radius of the circle using the following formula:

chord length = `2sqrt(r^2 - d^2)`

r expresses the radius of circle

d expresses the perpendicular distance from the center of circle to the chord

You need to find the chord length, hence, you should convert the given equation of the circle into standard form such that:

`(x-h)^2 + (y-k)^2 = r^2`

You should complete the squares such that:

`x^2 + y^2 - 2x - 6y + 6 = 0`

`(x^2 - 2x + 1)+ (y^2 - 6y + 9) - 1 - 9 +6 = 0`

`(x-1)^2 + (y-3)^2 = 4`

You need to identify the center and radius of circle that has the equation `x^2 + y^2 - 2x - 6y + 6 = 0`  such that:

`C(1,3) and r = 2`

The length of the chord is the length of diameter, hence chord length = 4.

You need to find the perpendicular distance from the center (2,1) of circle to the chord of length 4, such that:

`d = |(2-1)^2 + (1-3)^2 - 4|/(sqrt(1+1))`

`d = 1/sqrt2 = sqrt2/2`

Substituting 4 for chord length and `sqrt2/2`  for d yields:

`4 = 2sqrt(r^2 - 1/2)`

You need to raise to square both side to remove the square root such that:

`4 = r^2 - 1/2 =gt r^2 = 4 + 1/2`

`r^2 = 9/2 =gt r =+- sqrt3/2`

`r = sqrt3/2`

Hence, evaluating the radius of circle of center (2,1), under given conditions, yields`r = sqrt3/2.`

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