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 of the circle in standard form, find the c

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

Key Concepts

Completing the Square

Completing the square is an algebraic technique used to transform a quadratic expression into a perfect square trinomial plus or minus a constant. This process is essential to rewrite the equation of a circle in a way that clearly exposes its center and radius. It involves grouping terms and adding and subtracting the appropriate value so that each quadratic in x or y becomes a squared binomial.

Standard Form of a Circle

The standard form of a circle is expressed as (x - h)² + (y - k)² = r². In this form, (h, k) represents the center of the circle, and r is its radius. Rewriting a general quadratic equation into this form simplifies the interpretation of the circle's geometry and makes the subsequent graphing process straightforward.

Center and Radius Identification

Once the equation is rewritten in standard form, the coordinates of the circle’s center can be directly read as the constant terms inside the squared expressions, and the radius is obtained by taking the square root of the constant on the right-hand side. This step is key to understanding the geometric properties of the circle and to accurately sketch its graph.

Graphing a Circle

Graphing a circle involves plotting its center, measuring the radius, and drawing a curve that is equidistant from the center at all points. This visualization step reinforces the understanding of the circle's geometry and helps in interpreting how the algebraic manipulation relates to the graphical representation.

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