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Chapter 1: Problem 2

Find the center and the radius of the given circle. Sketch its graph. \(x^{2}+y^{2}=9\)

Short Answer

Expert verified
The center is (0,0) and the radius is 3.

Step by step solution

01

Recognize the Standard Circle Equation

The equation of the circle given is in the standard form, which is \[x^2 + y^2 = r^2\]where \((h, k)\) is the center and \(r\) is the radius of the circle. Since there are no linear terms, the center of the circle is at the origin \((0, 0)\).
02

Identify the Radius of the Circle

The equation \(x^2 + y^2 = 9\) represents a circle with radius \(r\). By comparing it to the standard form equation, we find that \(r^2 = 9\), which implies that the radius \(r = \sqrt{9} = 3\).
03

Sketch the Circle

To sketch the circle, draw a coordinate plane and mark the center at the origin, \((0, 0)\). Then, draw a circle with a radius of 3 units around the origin. The circle should extend 3 units in all directions (left, right, up, down) from the origin.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radius of a Circle
The radius of a circle is the distance from its center to any point on its boundary. It is a key feature that determines the size of the circle. When you have the equation of a circle, such as in the form \(x^2 + y^2 = r^2\), the radius \(r\) is found by taking the square root of the constant on the right side of the equation. In our given example \(x^2 + y^2 = 9\), the constant is 9, so we calculate \(r = \sqrt{9} = 3\). This means each point on the edge of the circle is exactly 3 units away from the center.
Center of a Circle
The center of a circle is the point from which every point on the circle is equidistant. In the standard circle equation \((x-h)^2 + (y-k)^2 = r^2\), the center is represented by \((h, k)\). For our specific circle equation \(x^2 + y^2 = 9\), we notice there are no \(h\) or \(k\) terms, which simplifies the equation to \((x-0)^2 + (y-0)^2 = 9\). Thus, the center of the circle is at \((0, 0)\), the origin of the coordinate plane. This makes it straightforward to sketch, as it is centered precisely where the x-axis and y-axis intersect.
Sketching Circles
Sketching a circle on a graph is a useful skill that helps visualize mathematical equations in geometry. To sketch a circle:
  • Begin by drawing a coordinate plane with a horizontal x-axis and a vertical y-axis.
  • Identify the center of the circle. For our problem, the center is at \((0, 0)\).
  • Use the radius to measure distance from the center. Our radius is 3, so mark points 3 units away from the center in all directions (up, down, left, right).
  • Connect these points in a smooth, round curve to complete the circle.
This simple process helps you accurately sketch a circle based on its equation, providing a visual representation of the mathematical relationship.

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Most popular questions from this chapter

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