In Exercises 31–40, write the standard form of the equation of the circle with the given center and radius. Center (-1, 4), r = 2
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Identify the standard form of the equation of a circle, which is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
Substitute the given center \((-1, 4)\) into the equation, replacing \(h\) with \(-1\) and \(k\) with \(4\).
Substitute the given radius \(r = 2\) into the equation.
Write the equation as \((x - (-1))^2 + (y - 4)^2 = 2^2\).
Simplify the equation to \((x + 1)^2 + (y - 4)^2 = 4\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Form of a Circle's Equation
The standard form of a circle's equation is given by (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius. This format allows for easy identification of the circle's center and radius, making it fundamental for graphing and analyzing circles in coordinate geometry.
The center of a circle is represented by the coordinates (h, k). In this case, the center is given as (-1, 4), meaning the circle is located at the point where x = -1 and y = 4 on the Cartesian plane. Understanding how to interpret these coordinates is essential for correctly applying them in the standard form equation.
The radius of a circle is the distance from the center to any point on the circle. It is denoted by r and is crucial for determining the size of the circle. In this problem, the radius is given as 2, which means that the circle extends 2 units in all directions from its center, affecting the equation's right side in the standard form.