EQUATION OF A CIRCLE WHICH PASSES THROUGH ORIGIN AND GIVEN CENTER 

Find the equation of circle with the center and passes through the given point.

Problem 1 :

Center : (11, 0)

Point on Circle : (3, 0)

Solution :

Equation of a circle with center (h, k) and radius r :

(x - h)² + (y - k)² = r²

Centre (h, k) = (11, 0)

(x - 11)² + (y - 0)² = r² --- (1)

The given circle is passing through the point (3, 0).

Then substitute 3 for x and 0 for y.

(3 - 11)² + (0 - 0)² = r²

(-8)² = r²

64 = r²

Standard equation of a circle :

(x - 11)² + (y - 0)² = 64

x² + (11)² - 2(x) (11) + y² = 64

x² + 121 - 22x + y² = 64

x² + y² - 22x + 121 = 64

Subtract 64 from each side.

General form of equation of a circle :

x² + y² - 22x + 57 = 0

Problem 2 :

Center : (15, 13)

Point on Circle : (19, 13)

Solution :

Equation of a circle with centre (h, k) and radius r :

(x - h)² + (y - k)² = r²

Centre (h, k) = (15, 13)

(x - 15)² + (y - 13)² = r² --- (1)

The given circle is passing through the point (19, 13).

Then substitute 19 for x and 13 for y.

(19 - 15)² + (13 - 13)² = r²

(4)² = r²

16 = r²

Standard equation of a circle :

(x - 15)² + (y - 13)² = 16

x² + (15)² - 2(x) (15) + y² + (13)² - 2(y)(13) = 16

x² + 225 - 30x + y² + 169 - 26y = 16

x² + y² - 30x - 26y + 394 = 16

Subtract 16 from each side.

General form of equation of a circle :

x² + y² - 30x - 26y + 378 = 0

Problem 3 :

Center : (-5, 9)

Point on Circle : (-7, 11)

Solution :

Equation of a circle with centre (h, k) and radius r :

(x - h)² + (y - k)² = r²

Centre (h, k) = (-5, 9)

(x + 5)² + (y - 9)² = r² --- (1)

The given circle is passing through the point (-7, 11).

Then substitute -7 for x and 11 for y.

(-7 + 5)² + (11 - 9)² = r²

(-2)² + (8)² = r²

4 + 64 = r²

68 = r²

Standard equation of a circle :

(x + 5)² + (y - 9)² = 68

x² + (5)² + 2(x) (5) + y² + 9² - 2(y)(9) = 68

x² + 25 + 10x + y² + 81 - 18y = 68

x² + y² + 10x - 18y + 106 = 68

Subtract 68 from each side.

General form of equation of a circle :

x² + y² + 10x - 18y + 38 = 0

Problem 4 :

Center : (-11, 11)

Point on Circle : (-15, 17)

Solution :

Equation of a circle with centre (h, k) and radius r :

(x - h)² + (y - k)² = r²

Centre (h, k) = (-11, 11)

(x + 11)² + (y - 11)² = r² --- (1)

The given circle is passing through the point (-15, 17).

Then substitute -15 for x and 17 for y.

(-15 + 11)² + (17 - 11)² = r²

(-4)² + (6)² = r²

16 + 36 = r²

52 = r²

Standard equation of a circle :

(x + 11)² + (y - 11)² = 52

x² + (11)² + 2(x) (11) + y² + (11)² - 2(y)(11) = 52

x² + 121 + 22x + y² + 121 - 22y = 52

x² + y² + 22x - 22y + 242 = 52

Subtract 52 from each side.

General form of equation of a circle :

x² + y² + 22x - 22y + 190 = 0

Problem 5 :

Find the equation of a circle where the center is at (2, -4), and the point (6, 1) rests on the circle.

Solution :

Center : (2, -4)

Point on Circle : (6, 1)

Equation of a circle with center (h, k) and radius r :

(x - h)² + (y - k)² = r²

Centre (h, k) = (2, -4)

(x - 2)² + (y + 4)² = r² --- (1)

The given circle is passing through the point (6, 1).

Then substitute 6 for x and 1 for y.

(6 - 2)² + (1 + 4)² = r²

(4)² + (5)² = r²

16 + 25 = r²

41 = r²

Standard equation of a circle :

(x - 2)² + (y + 4)² = 41

x² + (2)² - 2(x) (2) + y² + (4)² + 2(y)(4) = 41

x² + 4 - 4x + y² + 16 + 8y = 41

x² + y² - 4x + 8y + 20 = 41

Subtract 41 from each side.

General form of equation of a circle : 

x² + y² - 4x + 8y - 21 = 0

Problem 6 :

Find the equation of a circle where the center is at (-2, 3), and the point (1, 4) rests on the circle.

Solution :

Center : (-2, 3)

Point on Circle : (1, 4)

Equation of a circle with centre (h, k) and radius r :

(x - h)² + (y - k)² = r²

Centre (h, k) = (-2, 3)

(x + 2)² + (y - 3)² = r² --- (1)

The given circle is passing through the point (1, 4).

Then substitute 1 for x and 4 for y.

(1 + 2)² + (4 - 3)² = r²

(3)² + (1)² = r²

9 + 1 = r²

10 = r²

Standard equation of a circle :

(x + 2)² + (y - 3)² = 10

x² + (2)² + 2(x) (2) + y² + (3)² - 2(y)(3) = 10

x² + 4 + 4x + y² + 9 - 6y = 10

x² + y² + 4x - 6y + 13 = 10

Subtract 10 from each side.

General form of equation of a circle : 

x² + y² + 4x - 6y + 3 = 0