# EQUATION OF A TRANSLATED CIRCLE FROM STANDARD FORM

Problem 1 :

(x - 16)2 + (y - 6)2 = 1

Translated 4 left, 2 up

Solution:

The standard form of a circle is

(x - h)2 + (y - k)2 = r2

Where r is the radius of the circle and (h, k) is the center of the circle.

Center (h, k) = (16, 6)

When the circle is translated left 4 units and up 2 units.

The center should be changed as,

((x - 16) - (-4))2 + ((y - 6) - (2))2 = 1

(x - 16 + 4)2 + (y - 6 - 2)2 = 1

(x - 12)2 + (y - 8)2 = 1

Problem 2 :

(x + 5)2 + (y + 7)2 = 36

Translated 5 left, 4 down

Solution:

The standard form of a circle is

(x - h)2 + (y - k)2 = r2

Where r is the radius of the circle and (h, k) is the center of the circle.

(x - (-5))2 + (y - (-7))2 = 36

Center (h, k) = (-5, -7)

Translating 5 units left and 4 units down

((x + 5) - (-5))2 + ((y + 7) - (-4))2 = 36

(x + 5 + 5)2 + (y + 7 + 4)2 = 36

(x + 10)2 + (y + 11)2 = 36

Problem 3 :

x2 + y2 + 14x + 12y + 76 = 0

Translated 2 right, 4 down

Solution:

x2 + y2 + 14x + 12y + 76 = 0

x2 + 14x + y2 + 12y + 76 = 0

x2 + 2 ⋅ x ⋅ 7 + 72 - 72 + y2 + 2 ⋅ y ⋅ 6 + 62 - 62 + 76 = 0

(x + 7)2 - 49 + (y + 6)2 - 36 + 76 = 0

(x + 7)2 + (y + 6)2 - 85 + 76 = 0

(x + 7)2 + (y + 6)2 = 9

Before translation :

Center of the circle is (7, -6).

(x + 7)2 + (y + 6)2 = 9

After translation :

(x + 7 - 2)2 + (y + 6 - (-4))2 = 9

Translated 2 right, 4 down

So, equation of new circle is,

(x + 5)2 + (y + 10)2 = 9

Problem 4 :

x2 + y2 - 10x + 20y + 61 = 0

Translated 1 left, 2 down

Solution:

x2 + y2 - 10x + 20y + 61 = 0

x2  - 2 ⋅ x ⋅ 5 + 52 - 52+ y2+ 2 ⋅ y ⋅ 10 + 102 - 102 + 61 = 0

(x - 5)2  - 25 + (y + 10)2 - 100 + 61 = 0

(x - 5)2 + (y + 10)2 - 25 - 100 + 61 = 0

(x - 5)2 + (y + 10)2 - 64 = 0

(x - 5)2 + (y + 10)2 = 64

Before translation, the center will be at :

(5, -10)

(x - 5)2 + (y + 10)2 = 64

After translation, the center will be at :

Translated 1 left, 2 down

(x - 5 - (-1))2 + ((y + 10) - (-2))2 = 64

(x - 5 + 1)2 + (y + 10 + 2)2 = 64

(x - 4)2 + ((y + 12)2 = 64

Problem 5 :

x2 + y2 + 14x - 8y + 29 = 0

Translated 3 right, 4 down

Solution:

x2 + y2 + 14x - 8y + 29 = 0

x2 + 14x + y2 - 8y + 29 = 0

x2 + 2⋅x⋅7 + 7- 72 + y2 - 2⋅y⋅4 + 42- 42 + 29 = 0

(x + 7)2 - 49 + (y - 4)2 - 16 + 29 = 0

(x + 7)2 + (y - 4)2 - 65 + 29 = 0

(x + 7)2 + (y - 4)2 - 36 = 0

(x + 7)2 + (y - 4)2 = 36

Before the translation, center will be at :

(-7, 4)

After the translation, center will be at :

(x + 7 - 3)2 + (y - 4 - (-4))2 = 36

(x + 4)2 + (y - 4 + 4)2 = 36

(x + 4)2 + y2 = 36

Problem 6 :

4y + y2 = -28x - x2 - 191

Translated 4 right

Solution:

4y + y2 = -28x - x2 - 191

x2 + 28x + y24y + 191 = 0

(x + 14)2 + (y - 2)2 - 142 - 22+ 191 = 0

(x + 14)2 + (y - 2)2 - 196  - 4 + 191 = 0

(x + 14)2 + (y - 2)2 - 200 + 191 = 0

(x + 14)2 + (y - 2)2 - 9 = 0

(x + 14)2 + (y - 2)2  = 9

Thus the coordinates of the center at (-14, -2). When the circle is translated right 4 units,

h = -14 + 4 = -10

So, equation of new circle is,

(x + 10)2 + (y + 2)2 = 9

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