In Exercises 53–64, complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. x² + y²+8x-2y-8=0

Verified step by step guidance

1. Step 1: Group the x and y terms together. Start with the equation: x^2 + 8x + y^2 - 2y - 8 = 0.
2. Step 2: Move the constant term to the other side of the equation: x^2 + 8x + y^2 - 2y = 8.
3. Step 3: Complete the square for the x terms. Take half of the coefficient of x, which is 8, divide by 2 to get 4, and square it to get 16. Add and subtract 16 inside the equation: (x^2 + 8x + 16) - 16.
4. Step 4: Complete the square for the y terms. Take half of the coefficient of y, which is -2, divide by 2 to get -1, and square it to get 1. Add and subtract 1 inside the equation: (y^2 - 2y + 1) - 1.
5. Step 5: Rewrite the equation in standard form by combining the completed squares and moving constants to the other side: (x + 4)^2 + (y - 1)^2 = 25. Identify the center as (-4, 1) and the radius as 5.

Key Concepts

Completing the Square

Completing the square is a method used to transform a quadratic equation into a perfect square trinomial. This technique involves rearranging the equation and adding or subtracting a constant to create a binomial squared. It is essential for converting equations into standard form, particularly for circles and parabolas, making it easier to identify key features such as the center and radius.

Standard Form of a Circle

The standard form of a circle's equation is given by (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius. This format allows for quick identification of the circle's center and radius, which are crucial for graphing the circle accurately. Understanding this form is vital for solving problems related to circles in coordinate geometry.

Graphing Circles

Graphing circles involves plotting points that satisfy the circle's equation on a coordinate plane. The center of the circle is marked at (h, k), and the radius r determines the distance from the center to any point on the circle. Knowing how to graph circles helps visualize their properties and relationships with other geometric figures, enhancing overall comprehension of the topic.