Use each graph to determine an equation of the circle in (a) center-radius form and (b) general form.

Verified step by step guidance

1. Identify the center of the circle by finding the midpoint of the diameter. The diameter endpoints are (1, 3) and (9, 3).
2. Calculate the midpoint: $(\frac{1+9}{2},\frac{3+3}{2})=(5,3)$. This is the center of the circle.
3. Determine the radius by calculating the distance from the center (5, 3) to one of the points on the circle, such as (5, 7).
4. Use the distance formula: $r=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$. Here, $r=\sqrt{(5-5{)}^2+(7-3{)}^2}$.
5. Write the equation of the circle in center-radius form: $(x-5{)}^2+(y-3{)}^2=r^2$. Then expand and simplify to get the general form: $x^2+y^2+Dx+Ey+F=0$.

Key Concepts

Circle Equation in Center-Radius Form

The center-radius form of a circle's equation is expressed 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 useful for quickly identifying the center and radius from the equation, allowing for easy graphing and understanding of the circle's properties.

General Form of a Circle's Equation

The general form of a circle's equation is given by x² + y² + Dx + Ey + F = 0, where D, E, and F are constants. This form can be derived from the center-radius form by expanding and rearranging the equation. It is often used in algebraic manipulations and can be converted back to center-radius form to find the circle's center and radius.

Finding the Center and Radius from a Graph

To determine the center and radius of a circle from its graph, identify the center point, which is equidistant from all points on the circle. The radius can be calculated by measuring the distance from the center to any point on the circumference. This visual approach aids in understanding the geometric properties of circles and their equations.