In Exercises 53-64, complete the square and write the equation in standard form. Then give the c

step 1 We need to complete the square and write the equation in standard form. Find the radius in the c

In Exercises 53-64, complete the square and write the...

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Key Concepts

Graphing a Circle

Graphing a circle involves plotting its center on a coordinate plane and drawing a curve that represents all points a fixed distance (the radius) from that center. This process visually communicates the symmetry and spatial characteristics of the circle.

Center and Radius

The center and radius are critical components in defining a circle’s geometry. The center indicates the fixed point equidistant from every point on the circle, while the radius is the constant distance from the center to any point on the perimeter. Identifying these values is essential for understanding and graphing the circle.

Standard Form of a Circle

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) represents the center and r is the radius. Converting a general quadratic equation into this form through completing the square makes it straightforward to determine the circle’s location and size.

Completing the Square

Completing the square is an algebraic method used to transform quadratic equations into a perfect square plus a constant. This technique simplifies the process of rewriting equations, particularly for conic sections, allowing for easier identification of their geometric properties.

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