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

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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).

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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.

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06:03

Equations of Circles & Ellipses

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.

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05:32

Intro to Polar Coordinates

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.

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