FIND THE RADIUS OF THE CIRCLE PASSING THROUGH POINTS

Let the center be O(x, y). The given three points can be considered as A, B and C.

The distance between OA, OB and OC will be the same because they are radii. Using this, we can find the center. 

By finding the distance between center and point on the circumference of the circle, we can find radius.

Problem 1 :

Find the coordinates of the center of a circle passing through the points A(2, 1), B(5, -8) and C(2, -9). Also find the radius of this circle.

Solution:

Let the center point be O(a, b)

Distance between

OA = OB = OC (radii of circle)

OA=(a-2)2+(b-1)2OB=(a-5)2+(b+8)2OC=(a-2)2+(b+9)2OA=OB(a-2)2+(b-1)2=(a-5)2+(b+8)2(a-2)2+(b-1)2=(a-5)2+(b+8)2a2-4a+4+b2-2b+1=a2-10a+25+b2+16b+646a-18b-84=06(a-3b-14)=0a-3b-14=0a-3b=14→(1)OA=OC(a-2)2+(b-1)2=(a-2)2+(b+9)2(a-2)2+(b-1)2=(a-2)2+(b+9)2a2-4a+4+b2-2b+1=a2-4a+4+b2+18b+81 -2b-18b+1-81=0-2b-18b-80=02b+18b+80=020b=-80b=-8020b=-4

Applying the value of b in (1), we get

a - 3(-4) = 14

a + 12 = 14

a = 2

So, the center is O(2, -4).

Distance between O and A=( 2-2)2+(-4-1)2=0+(-5)2=25Radius of the circle=5

Problem 2 :

Find the coordinates of the center of a circle passing through the points A(-2, -3), B(-1, 0) and C(7, -6). Also find the radius of this circle.

Solution:

Let the center point be O(a, b)

Distance between

OA = OB = OC (radii of circle)

OA=(a+2)2+(b+3)2OB=(a+1)2+(b-0)2OC=(a-7)2+(b+6)2OA=OB(a+2)2+(b+3)2=(a+1)2+(b-0)2(a+2)2+(b+3)2=(a+1)2+(b-0)2a2+4a+4+b2+6b+9=a2+2a+1+b22a+6b+12=02(a+3b+6)=0a+3b+6=0a+3b=-6→(1)OB=OC(a+1)2+(b-0)2=(a-7)2+(b+6)2(a+1)2+(b-0)2=(a-7)2+(b+6)2a2+2a+1+b2=a2-14a+49+b2+12b+36 16a-12b-84=04(4a-3b-21)=04a-3b-21=04a-3b=21→(2)

(1) + (2)

(a + 3b) + (4a - 3b) = -6 + 21

5a = 15

a = 3

Applying the value of a in (1), we get

3 + 3b = -6

3b = -9

b = -3

So, the center is O(3, -3).

Distance between O and A=(3+2)2+(-3+3)2=52+0=25Radius of the circle=5

Problem 3 :

Find the coordinates of the center of a circle passing through the points A(1, 2), B(3, -4) and C(5, -6). Also find the radius of this circle.

Solution:

Let the center point be O(a, b)

Distance between

OA = OB = OC (radii of circle)

OA=(a-1)2+(b-2)2OB=(a-3)2+(b+4)2OC=(a-5)2+(b+6)2OA=OB(a-1)2+(b-2)2=(a-3)2+(b+4)2(a-1)2+(b-2)2=(a-3)2+(b+4)2a2-2a+1+b2-4b+4=a2-6a+9+b2+8b+164a-12b-20=04(a-3b-5)=0a-3b-5=0a-3b=5→(1)OB=OC(a-3)2+(b+4)2=(a-5)2+(b+6)2(a-3)2+(b+4)2=(a-5)2+(b+6)2a2-6a+9+b2+8b+16=a2-10a+25+b2+12b+36 4a-4b-36=04(a-b-9)=0a-b-9=0a-b=9→(2)

(1) - (2)

(a - 3b) - (a - b) = 5 - 9

-3b + b = -4

-2b = -4

b = 2

Applying the value of b in (1), we get

a - 3(2) = 5

a - 6 = 5

a = 11

So, the center is O(11, 2).

Distance between O and A=(11-1)2+(2-2)2=102+0=100Radius of the circle=10

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