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 (5, -4), radius 7

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Identify the center and radius of the circle. Here, the center is given as \((5, -4)\) and the radius is \(7\).

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

Substitute the given center coordinates and radius into the formula: \[(x - 5)^2 + (y - (-4))^2 = 7^2\].

Simplify the equation by replacing \(y - (-4)\) with \(y + 4\) and squaring the radius: \[(x - 5)^2 + (y + 4)^2 = 49\].

To graph the circle, plot the center at \((5, -4)\) on the coordinate plane, then draw a circle with radius \(7\) units around this point.

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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 shows the circle's location and size, making it easier to graph and analyze.

Understanding Coordinates of the Center

The center of the circle is given as a point (h, k) on the coordinate plane. Knowing the center helps position the circle correctly, as the equation uses these values to measure distances from this fixed point.

Graphing a Circle on the Coordinate Plane

Graphing involves plotting the center point and using the radius to mark points at equal distances in all directions. Connecting these points smoothly forms the circle, illustrating its size and position visually.

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