In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. See Examples 5 and 6.
center (-2, 5), radius 4

Verified step by step guidance

Recall the center-radius form of a circle's equation: \(\left(x - h\right)^2 + \left(y - k\right)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Identify the center coordinates and radius from the problem: center \((-2, 5)\) means \(h = -2\) and \(k = 5\), and radius \(r = 4\).

Substitute the values into the center-radius form: \(\left(x - (-2)\right)^2 + \left(y - 5\right)^2 = 4^2\).

Simplify the equation: \(\left(x + 2\right)^2 + \left(y - 5\right)^2 = 16\).

To graph the circle, plot the center at \((-2, 5)\), then draw a circle with radius 4 units around this point, marking points 4 units away in all directions (up, down, left, right).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Equation of a Circle in Center-Radius Form

The center-radius 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. This form directly relates the geometric properties of the circle to its algebraic representation.

Understanding Coordinates of the Center

The center of the circle is given as a point (h, k) in the coordinate plane. Identifying these coordinates correctly is essential for writing the equation and plotting the circle accurately.

Graphing a Circle on the Coordinate Plane

Graphing involves plotting the center point and using the radius to mark points at a fixed distance in all directions. Connecting these points smoothly forms the circle, helping visualize its size and position.

Recommended video: