Question

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

Accepted Answer

Identify the center of the circle by finding the midpoint between two opposite points on the circle. For example, use points (1, 3) and (9, 3). The midpoint formula is $$\left( \frac{x_1$ + x_2$}{2}, \frac{y_1$ + y_2$}{2} \right). $$Calculate the center as $$\left( \frac{1 + 9}{2}, \frac{3 + 3}{2} \right). $$Calculate the radius of the circle by finding the distance from the center to any point on the circle. Use the distance formula $$r = \sqrt{(x - h)^2$ + (y - k)^2$}$$, where $$(h, k) is $$the center and $$(x, y) is a $$point on the circle, for example (5, 7). Write the equation of the circle in center-radius form: $$(x - h)^2 + (y - k)^2 = r^2$$ where $$(h, k) is $$the center and $$r is $$the radius. Expand the center-radius form equation by squaring the binomials and simplifying to get the general form of the circle equation: $$x^2 + y^2 + Dx + Ey + F = 0. $$Group like terms and write the final general form equation by moving all terms to one side of the equation and simplifying coefficients.

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

Verified video answer for a similar problem:

Key Concepts

Center-Radius Form of a Circle

Finding the Center and Radius from a Graph

General Form of a Circle's Equation