Equation of Circle Formula: Definition, Forms, Solved Examples
Equation of Circle Formula: The equation of a circle in standard form is (x - a)2 + (y - b)2 = r2, where (a, b) is the center and r is the radius. In general form, it's x2 + y2 + Ax + By + C = 0, with A, B, and C describing the circle's position and size.
Manoj Kumar18 Sept, 2023
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Equation of Circle Formula: In mathematics, the circle equation formula represents the equation of a circle, describing its center-radius form. A circle is defined as a closed shape whose boundary consists of points that are equidistant from its center. To mathematically represent a circle, there are two primary forms:
Also Check - Factors and multiples Formula1. Standard Form of Circle Equation
The standard circle equation is expressed as follows: (x - a) 2 + (y - b) 2 = r 2 Here: (a, b) are the coordinates of the circle's center. "r" represents the circle's radius.2. General Form of Circle Equation
The general form of a circle equation is expressed as: x 2 + y 2 + Ax + By + C = 0 Also Check - Volume of a Cube FormulaEquation of Circle Formula Solved Examples
Question 1.What is the circle equation for a circle with a center at (4, 5) and a radius of 7 units? Solution: The provided parameters are as follows: Center (a, b) = (4, 5) Radius r = 7 The standard form of a circle equation expressed as follow: (x − a)² + (y − b)² = r² So, substituting the values, we have: (x − 4)² + (y − 5)² = 7² This simplifies to: (x − 4)² + (y − 5)² = 49 Also Check - Area of a Rectangle Formula Question 2. What is the centre and radius of the circle defined by the equation x² + y² – 10x + 14y + 38 = 0? Solution: The given circle equation is: x² + y² – 10x + 14y + 38 = 0 Rearrange it as follows: x² - 10x + y² + 14y + 38 = 0 Simplify both the x and y terms by completing the square for each. (x² - 10x + 25) + (y² + 14y + 49) - 25 - 49 = 0 Now, rewrite it as: (x - 5)² + (y + 7)² - 74 = 0 (x - 5)² + (y + 7)² = 74 Comparing this with the standard form (x - a)² + (y - b)² = r², we get: a = 5, b = -7, and r² = 74 Therefore, the center is (5, -7), and the radius is √74 units. Question 3. Express the general form of the circle equation with a center at (2, 3) and a radius of 1 unit. Solution: Given values: Center = (a, b) = (2, 3) Radius r = 1 The standard form of the circle equation expressed as follows: (x - a)² + (y - b)² = r² Put the values of the center and radius: (x - 2)² + (y - 3)² = 1² Simplify: x² - 4x + 4 + y² - 6y + 9 = 1 x² + y² - 4x - 6y + 13 - 1 = 0 x² + y² - 4x - 6y + 12 = 0 This is in the general form of a circle equation, where A = -4, B = -6, C = 9, Hence, the general form of the circle equation is x 2 + y 2 – 4x – 6y + 9 = 0. Question 4. Express the general form of the circle equation with a center at (2, 4) and a radius of 1 unit. Solution: Given values: Center = (a, b) = (2, 4) Radius r = 1 The standard form of the circle equation expressed as follows: (x - a)² + (y - b)² = r² Put the values of the center and radius: (x - 2)² + (y - 4)² = 1² Simplify: x² - 4x + 4 + y² - 8y + 16 = 1 x² + y² - 4x - 8y + 20 - 1 = 0 x² + y² - 4x - 8y + 19 = 0 This is in the general form of a circle equation, where A = -4, B = -8, C = 19, Hence, the general form of the circle equation is x 2 + y 2 – 4x – 6y + 19 = 0. Question 5. Given the center point and radius of a circle as (4, 6) and 9, respectively, represent this as a circle equation. Solution: The provided parameters are as follows: Center (a, b) = (4, 6) Radius r = 9 The standard form of a circle equation expressed as follows: (x − a)² + (y − b)² = r² So, Put the values, we have: (x − 4)² + (y − 6)² = 9² This simplifies to: (x − 4)² + (y − 6)² = 81Equation of Circle Formula FAQs
What is the equation of a circle in standard form?
The standard form for the equation of a circle is given as (x - a)^2 + (y - b)^2 = r^2, with (a, b) denoting the circle's center, and r representing its radius.
How is the general form of a circle equation expressed?
In the general form of a circle equation, it is expressed as x^2 + y^2 + Ax + By + C = 0, where A, B, and C serve as coefficients that provide information about the circle's location and dimensions.
What is the significance of the parameters (a, b) and r in the standard form of a circle equation?
(a, b) denote the coordinates of the circle's center, while r signifies the circle's radius.
How do you convert a standard form circle equation into a general form equation?
To convert a standard form equation to a general form equation, you expand and rearrange the terms to collect x and y variables on one side and constants on the other.
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