# 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

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