# CONVERT FROM GENERAL FORM TO STANDARD FORM OF A CIRCLE

Write each of the following in standard form. Identify the center of the circle as well as the length of the radius, the graph the circle if possible.

Problem 1 :

x2 + y2 + 10x - 4y + 13 = 0

Solution :

x2 + y2 + 10x - 4y + 13 = 0

Write x - terms and y - terms together.

x2 + 10x + y2 - 4y + 13 = 0

Completing the square.

x2 + 10x + 52 - 52 + y2 - 4y + 22 - 22 + 13 = 0

(x + 5)2 - 52 + (y - 2)2 - 22 + 13 = 0

(x + 5)2 - 25 + (y - 2)2 - 4 + 13 = 0

(x + 5)2 + (y - 2)2 - 16 = 0

(x + 5)2 + (y - 2)2  = 16

(x + 5)2 + (y - 2)2  = 42

The above equation of the circle is in standard form.

Center (h, k) = (-5, 2)

Problem 2 :

x2 + y2 + 6y - 16 = 0

Solution :

x2 + y2 + 6y - 16 = 0

Write x - terms and y - terms together.

x2 + y2 + 6y - 16 = 0

Completing the square.

x2 + y2 + 6y + 32 - 32 - 16 = 0

x2 + (y + 3)2 - 32 - 16 = 0

x2 + (y + 3)2 - 9 - 16 = 0

x2 + (y + 3)2 - 25 = 0

x2 + (y + 3)2  = 25

x2 + (y + 3)2  = 52

The above equation of the circle is in standard form.

Center (h, k) = (0, -3)

Problem 3 :

x2 + y2 - 4x - 8y + 11 = 0

Solution :

x2 + y2 - 4x - 8y + 11 = 0

Write x - terms and y - terms together.

x2 - 4x + y2 - 8y + 11 = 0

Completing the square.

x2 - 4x + 22 - 22 + y2 - 8y + 42 - 42 + 11 = 0

(x - 2)2 - 22 + (y - 4)2 - 42 + 11 = 0

(x - 2)2 - 4 + (y - 4)2 - 16 + 11 = 0

(x - 2)2 + (y - 4)2 - 9 = 0

(x - 2)2 + (y - 4)2 = 9

(x - 2)2 + (y - 4)2 = 32

The above equation of the circle is in standard form.

Center (h, k) = (2, 4)

Problem 4 :

x2 + y2 + 6x - 8y + 29 = 0

Solution :

x2 + y2 + 6x - 8y + 29 = 0

Write x - terms and y - terms together.

x2 + 6x + y2 - 8y + 29 = 0

Completing the square.

x2 + 6x + 32 - 32 + y2 - 8y + 42 - 42 + 29 = 0

(x + 3)2 - 32 + (y - 4)2 - 42 + 29 = 0

(x + 3)2 - 9 + (y - 4)2 - 16 + 29 = 0

(x + 3)2 + (y - 4)2 + 4 = 0

Subtract 4 on each sides.

(x + 3)2 + (y - 4)2 = -4

There is no circle.

Problem 5 :

x2 + y2 - 8x - 20 = 0

Solution :

x2 + y2 - 8x - 20 = 0

Write x - terms and y - terms together.

x2 - 8x + y2 - 20 = 0

Completing the square.

x2 - 8x + 42 - 42 + y2 - 20 = 0

(x - 4)2 - 42 + y2 - 20 = 0

(x - 4)2 - 16 + y2 - 20 = 0

(x - 4)2 + y2 - 36 = 0

(x - 4)2 + y2 = 36

(x - 4)2 + y2 = 6

The above equation of the circle is in standard form.

Center (h, k) = (4, 0)

Problem 6 :

2x2 + 2y2 + 4x - 12y + 2 = 0

Solution :

2x2 + 2y2 + 4x - 12y + 2 = 0

2(x2 + y2 + 2x - 6y + 1) = 0

Dividing 2 on each sides.

x2 + y2 + 2x - 6y + 1 = 0

Write x - terms and y - terms together.

x2 + 2x + y2 - 6y + 1 = 0

Completing the square.

x2 + 2x + 12 - 12 + y2 - 6y  + 32 - 32  + 1 = 0

(x + 1)2 - 12 + (y - 3)2 - 32 + 1 = 0

(x + 1)2 - 1 + (y - 3)2 - 9 + 1 = 0

(x + 1)2 + (y - 3)2 - 9 = 0

(x + 1)2 + (y - 3)2 = 9

(x + 1)2 + (y - 3)2 = 32

The above equation of the circle is in standard form.

Center (h, k) = (-1, 3)

Problem 7 :

3x2 + 3y2 + 36x + 30y + 171 = 0

Solution :

3x2 + 3y2 + 36x + 30y + 171 = 0

3(x2 + y2 + 12x + 10y + 57) = 0

Dividing 3 on each sides.

x2 + y2 + 12x + 10y + 57 = 0

Write x - terms and y - terms together.

x2 + 12x + y2 + 10y + 57 = 0

Completing the square.

x2 + 12x + 62 - 62 + y2 + 10y + 52 - 52 + 57 = 0

(x + 6)2 - 62 + (y + 5)2 - 52 + 57 = 0

(x + 6)2 - 36 + (y + 5)2 - 25 + 57 = 0

(x + 6)2 + (y + 5)2 - 4 = 0

(x + 6)2 + (y + 5)2 = 4

(x + 6)2 + (y + 5)2 = 22

The above equation of the circle is in standard form.

Center (h, k) = (-6, -5)

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