Use each graph to determine an equation of the circle in (a) center-radius form and (b) general form.

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Identify the center of the circle by finding the midpoint between two opposite points on the circle. For example, use points (1, 3) and (9, 3). The midpoint formula is \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\). Calculate the center as \(\left( \frac{1 + 9}{2}, \frac{3 + 3}{2} \right)\).

Calculate the radius of the circle by finding the distance from the center to any point on the circle. Use the distance formula \(r = \sqrt{(x - h)^2 + (y - k)^2}\), where \((h, k)\) is the center and \((x, y)\) is a point on the circle, for example (5, 7).

Write the equation of the circle in center-radius form: \[(x - h)^2 + (y - k)^2 = r^2\] where \((h, k)\) is the center and \(r\) is the radius.

Expand the center-radius form equation by squaring the binomials and simplifying to get the general form of the circle equation: \[x^2 + y^2 + Dx + Ey + F = 0\].

Group like terms and write the final general form equation by moving all terms to one side of the equation and simplifying coefficients.

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주요 개념

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Center-Radius Form of a Circle

The center-radius form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. This form directly shows the circle's center and radius, making it easy to graph or identify key features.

Finding the Center and Radius from a Graph

To find the center, identify the midpoint of the diameter points on the circle. The radius is the distance from the center to any point on the circle. Using the distance formula helps calculate the radius accurately.

General Form of a Circle's Equation

The general form is x² + y² + Dx + Ey + F = 0, derived by expanding the center-radius form and simplifying. It is useful for algebraic manipulation and solving systems but less intuitive for identifying the circle's center and radius.