The equation of a circle captures its geometric essence, defining a set of points equidistant from a center. Learn how to derive and use the standard form \\(x-h)^2 + (y-k)^2 = r^2\\ when the center and radius are known, and the general form \\(x^2 + y^2 + Ax + By + C = 0\\) for expanded equations. Understand how to locate points in relation to a circle and the criteria for a valid circle equation.
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A circle is a collection of points in a plane that are all at the same distance from a central point
Cartesian Coordinate System
The standard equation of a circle in a Cartesian coordinate system is \((x-h)^2 + (y-k)^2 = r^2\)
Known Center and Radius
The standard equation of a circle can be easily written when the center and radius are known
The general form of a circle's equation is \(x^2 + y^2 + Ax + By + C = 0\), where A, B, and C are constants
The coefficients of \(x^2\) and \(y^2\) in a circle's equation must be equal and non-zero
The discriminant \((A/2)^2 + (B/2)^2 - C\) must be positive for a real and positive radius
A minimum of three non-collinear points are necessary to uniquely determine a circle
The equation of a circle centered at the origin is \(x^2 + y^2 = r^2\)
The radius of a circle can be determined using the distance formula when the center and a point on the circle are known
The position of a point relative to a circle can be determined by plugging its coordinates into the circle's equation
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Definition of a circle
A set of points equidistant from a central point in a plane.
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Circle's radius significance
Constant distance from circle's center to any point on circumference.
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Circle equation derivation basis
Derived from Pythagorean theorem, ensuring equidistance of radius.
