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Equation of a Circle

The equation of a circle maps the circle on a two-dimensional coordinate plane. Following are some useful forms of the equation of the circle.
 

Standard Form

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

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

Polar Form 

x2+y= r2

Where x=rcosθ and y=rsinθ.

General Form 

x2+y2+Ax+By+C = 0

Where A, B, and C are constant.

 

Equation of a Circle Sample Questions:

Question 1: What would be the center of the following equation of the circle?


(x-2)2+(y-3)= 62


Answer: (2,3)

Explanation: Use the formula of standard form of the equation of a circle.

 

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Question 2: What would be the radius of the following equation of the circle?


(x-5)2+(y-7)= 49


Answer: 7 units

Explanation: Use the formula of standard form of the equation of a circle.

 

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Question 3: Write the equation of circle in standard form for the following figure:


Graph 4


Answer:  (x-3)2+(y-2)= 32

Explanation: The given figure has the center as 3,2 and radius as r=3 units.
Now, let us plug those values, we get.

 

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Question 4: Identify the center and radius of the circle, and sketch its graph:


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


Graph 2


Answer: Graph 3

Explanation: Use the formula of standard form of the equation of a circle.

 

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Question 5: Translate the following equation of circle by 4 units right, 2 units down  (x+3)2+(y-5)= 25.


Answer: (x+7)2+(y-3)= 25

Explanation: Translate the 4 units down and then translate 2 units down.

 

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Question 6: Translate the following equation of circle by 3 units left, 5 units up  (x+3)2+(y-5)= 64.

Answer: (x)2+(y)= 64

Explanation: Translate the 3 units down and then translate 5 units up.

 

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Question 7: Identify the center and radius of the circle, and sketch its graph:


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


Graph


Answer: Graph 1


Center is at 2,-3 and radius as r=3 units

Explanation: Use the formula of standard form of the equation of a circle.

 

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Question 8: Write the following equation of the circle in standard form


x2+ 8x - 2y = 64 - y2


Answer: (x+4)2+(y+1)= 81

Explanation: Rewrite the equation and complete the square.

 

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Question 9: Write the following equation of the circle in polar form.


x2+ 7x + y2 - 5y = 0

 

Answer: r=-8cosθ + 2sinθ

Explanation: Plug x+ y= r2, and x=rcosθ, y=rsinθ

 

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Question 10: Write the following equation of the circle in polar form.


x- 9x + y2 + 5y = 0


Answer: r=9cosθ - 5sinθ

Explanation: Plug x+ y= r2, and x=rcosθ, y=rsinθ

 

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