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 – 10x – 6y – 30 = 0

Verified step by step guidance

1. Start by rearranging the equation to group the x and y terms: \(x^2 - 10x + y^2 - 6y = 30\).
2. To complete the square for the x terms, take half of the coefficient of x, which is -10, divide by 2 to get -5, and then square it to get 25. Add and subtract 25 inside the equation: \((x^2 - 10x + 25) - 25\).
3. Similarly, complete the square for the y terms. Take half of the coefficient of y, which is -6, divide by 2 to get -3, and then square it to get 9. Add and subtract 9 inside the equation: \((y^2 - 6y + 9) - 9\).
4. Rewrite the equation with the completed squares: \((x - 5)^2 - 25 + (y - 3)^2 - 9 = 30\).
5. Simplify the equation to standard form: \((x - 5)^2 + (y - 3)^2 = 64\). The center of the circle is \((5, 3)\) and the radius is \(\sqrt{64} = 8\).

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 constants 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) represents 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 interpreting and manipulating circle equations.

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.