Work each of the following. Find the equation of a circle with center at (-4, 3), passing through the point (5, 8).Write it in center-radius form.

Verified step by step guidance

1. insert step 1: Recall the center-radius form of a circle's equation, which is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius.
2. insert step 2: Identify the center of the circle from the problem, which is given as (-4, 3). Therefore, h = -4 and k = 3.
3. insert step 3: Use the distance formula to find the radius r. The distance formula is r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}, where (x_1, y_1) is the center and (x_2, y_2) is the point on the circle.
4. insert step 4: Substitute the center (-4, 3) and the point (5, 8) into the distance formula to calculate r.
5. insert step 5: Substitute the values of h, k, and r into the center-radius form equation to write the equation of the circle.

Key Concepts

Equation of a Circle

The equation of a circle in center-radius form 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 formula allows us to describe all points (x, y) that are a fixed distance (r) from the center.

Distance Formula

The distance formula is used to calculate the distance between two points in a coordinate plane, given by the formula d = √((x₂ - x₁)² + (y₂ - y₁)²). In the context of circles, this formula helps determine the radius by measuring the distance from the center to any point on the circle.

Center-Radius Form

Center-radius form is a specific way to represent the equation of a circle that highlights its center and radius. This form is particularly useful for graphing and understanding the geometric properties of the circle, as it directly relates the circle's position to its size.