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 center of the r

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

Key Concepts

Completing the Square

Completing the square is a technique used to transform a quadratic expression into a perfect square trinomial. This method involves adding and subtracting the appropriate constant to rewrite the expression as the square of a binomial, which is essential when converting equations of circles or parabolas into their standard forms.

Standard Form of a Circle

The standard form of a circle's equation is expressed as (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r is the radius. This form is particularly useful because it directly reveals the geometric properties of the circle, such as its location and size, making further analysis and graphing straightforward.

Identifying the Center and Radius

Once the equation of a circle is written in standard form, the center and radius can be identified by directly comparing the equation to (x - h)² + (y - k)² = r². The coordinates (h, k) indicate the circle's center, while the radius r is the square root of the constant term on the right-hand side. This step is fundamental for understanding the geometry and position of the circle.

Graphing a Circle

Graphing a circle involves plotting the circle on a coordinate plane using its center and radius. After identifying the center and the radius, one can mark the center point, and then use the radius to determine several points around the perimeter. Drawing a smooth curve through these points completes the sketch, accurately reflecting the circle's geometry.

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