Describe the graph of each equation as a circle, a point, or nonexistent. If it is a circle, give the center and radius. If it is a point, give the coordinates. See Examples 3–5. x^2+y^2+4x+14y=-54

Verified step by step guidance

1. Rewrite the given equation in the standard form of a circle by completing the square for both x and y terms.
2. Start with the x terms: x^2 + 4x. To complete the square, take half of the coefficient of x, which is 4, divide by 2 to get 2, and then square it to get 4. Add and subtract 4 inside the equation.
3. Next, complete the square for the y terms: y^2 + 14y. Take half of the coefficient of y, which is 14, divide by 2 to get 7, and then square it to get 49. Add and subtract 49 inside the equation.
4. Rewrite the equation incorporating the completed squares: (x + 2)^2 + (y + 7)^2 = -54 + 4 + 49.
5. Simplify the right side of the equation to determine if it represents a circle, a point, or is nonexistent.

Key Concepts

Standard Form of a Circle

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. To identify a circle from a general equation, it is often necessary to rearrange the equation into this standard form by completing the square for both x and y terms.

Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This technique involves taking half of the coefficient of the linear term, squaring it, and adding it to both sides of the equation. This is essential for rewriting the equation of a circle in standard form.

Identifying Nonexistent Solutions

In some cases, the equation may not represent a circle or point, particularly if the resulting radius is negative or if the equation simplifies to an inconsistency. Understanding how to analyze the final form of the equation helps determine whether the graph is a circle, a single point, or nonexistent.