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)2 = r2 Where h,k is the coordinate of the center of the circle, and r is the radius. |
Polar Form | x2+y2 = 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)2 = 62
Answer: (2,3)
Explanation: Use the formula of standard form of the equation of a circle.
Question 2: What would be the radius of the following equation of the circle?
(x-5)2+(y-7)2 = 49
Answer: 7 units
Explanation: Use the formula of standard form of the equation of a circle.
Question 3: Write the equation of circle in standard form for the following figure:
Answer: (x-3)2+(y-2)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.
Question 4: Identify the center and radius of the circle, and sketch its graph:
(x+1)2+(y-2)2 = 16
Answer:
Explanation: Use the formula of standard form of the equation of a circle.
Question 5: Translate the following equation of circle by 4 units right, 2 units down (x+3)2+(y-5)2 = 25.
Answer: (x+7)2+(y-3)2 = 25
Explanation: Translate the 4 units down and then translate 2 units down.
Question 6: Translate the following equation of circle by 3 units left, 5 units up (x+3)2+(y-5)2 = 64.
Answer: (x)2+(y)2 = 64
Explanation: Translate the 3 units down and then translate 5 units up.
Question 7: Identify the center and radius of the circle, and sketch its graph:
(x-2)2+(y+3)2 = 9
Answer:
Center is at 2,-3 and radius as r=3 units
Explanation: Use the formula of standard form of the equation of a circle.
Question 8: Write the following equation of the circle in standard form
x2+ 8x - 2y = 64 - y2
Answer: (x+4)2+(y+1)2 = 81
Explanation: Rewrite the equation and complete the square.
Question 9: Write the following equation of the circle in polar form.
x2+ 7x + y2 - 5y = 0
Answer: r=-8cosθ + 2sinθ
Explanation: Plug x2 + y2 = r2, and x=rcosθ, y=rsinθ
Question 10: Write the following equation of the circle in polar form.
x2 - 9x + y2 + 5y = 0
Answer: r=9cosθ - 5sinθ
Explanation: Plug x2 + y2 = r2, and x=rcosθ, y=rsinθ
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