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, write the equation in standard form of the circle, and then find

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

Key Concepts

Completing the Square

Completing the square is an algebraic method used to convert a quadratic equation into a perfect square trinomial plus a constant. This method makes it easier to rewrite equations in a form that reveals important properties such as the vertex in parabolas or the center in circles. It involves grouping the x and y terms, adding and subtracting appropriate constants, and reorganizing the equation into squared binomials.

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) is the center of the circle and r is the radius. This form is particularly useful because it immediately presents the geometric characteristics of the circle, making further analysis and graphing straightforward.

Center and Radius of a Circle

Once the equation of a circle is rewritten in standard form, the center and radius can be directly identified. The numbers h and k represent the coordinates of the center, while the square root of the constant on the right-hand side gives the radius. These values are essential for understanding the circle's location and size in a coordinate plane.

Graphing the Equation of a Circle

Graphing a circle using its standard form involves plotting the center at (h, k) and then drawing a circle with the radius r around that center. This visual representation helps to understand the circle’s spatial properties, showing how it is oriented and scaled in the coordinate system.

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