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 to write the equation of the circle in standard form, find the ce

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

Key Concepts

Center and Radius Identification

Once a circle’s equation is rewritten in standard form, the coordinates of the circle’s center are provided by h and k, and the radius is determined by taking the square root of the constant on the right side of the equation. This identification simplifies the process of graphing the circle and understanding its spatial properties.

Graphing Conic Sections

Graphing a circle involves plotting its center on a coordinate plane and drawing a circle around that point with the radius determined from its equation. This process is a part of a broader set of techniques in graphing conic sections, where understanding and using the standard form makes it easier to visualize the shape and position of the graph.

Completing the Square

This is an algebraic technique used to transform a quadratic expression into a perfect square form. By adding and subtracting appropriate constant terms, the quadratic expressions for x and y can be rewritten as squared binomials, a critical step in rewriting conic sections like circles into a more interpretable form.

Standard Form of a Circle

The standard form of a circle’s equation is written as (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r is its radius. This form is beneficial because it directly reveals the circle’s key geometric features, facilitating further analysis and graphing.

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