Textbook Question
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 (0, 0), radius 6
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0. Review of College Algebra
Basics of Graphing
3:10 minutesProblem 59
Textbook Question
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
Verified step by step guidance1
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.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
3mKey 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.
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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.
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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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