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Chapter 10: Problem 146

Graph each circle. Identify the center and the radius. \(x^{2}+y^{2}=25\)

Short Answer

Expert verified
Center: (0, 0). Radius: 5.

Step by step solution

01

- Understand the Equation Form

Recognize that the given equation \(x^2 + y^2 = 25\) is in the standard form of a circle equation \(x^2 + y^2 = r^2\).
02

- Identify the Radius

Compare the given equation \(x^2 + y^2 = 25\) with the standard form \(x^2 + y^2 = r^2\). We see that \(r^2 = 25\), so the radius \(r\) is \(5\) (since \(r = \sqrt{25}\)).
03

- Identify the Center

Since there is no \(x\) or \(y\) term being subtracted or added, the circle is centered at the origin, \((0, 0)\).
04

- Graph the Circle

Plot the center of the circle at \((0, 0)\) on the graph. Using the radius \(r = 5\), plot points 5 units away from the center in all directions (up, down, left, right). Then, draw the circle passing through these points.

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Key Concepts

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

graphing circles
Graphing circles can be a fun way to visualize and understand the concept of circular shapes in mathematics. The key to graphing a circle lies in its equation. The general form of a circle's equation in Cartesian coordinates is: isn't necessarily a point-by-point plotting method, it's about recognizing the center and radius from the equation and using them to define the boundary of the circle on a graph. To graph a circle, follow these basic steps:
  • Identify the center of the circle.
  • Determine the radius.
  • Plot the center point on the graph.
  • Measure and mark points equal to the radius distance in all directions (up, down, left, right).
  • Connect these points smoothly to complete the circle.

radius of a circle
The radius of a circle is the distance from the center to any point on the circle's edge. It's a fundamental property that helps define the size of the circle. In the equation of a circle in standard form ou have:
  • The squared term represents the radius squared.
  • By finding the square root of this number, you can determine the actual radius.



This means the radius is



In summary, knowing the radius allows you to accurately graph the circle and understand its size.
center of a circle
The center of the circle is the point from which all points on the circle are equidistant. Essentially, it's the 'heart' of the circle. In the standard equation of the circle, the center can be found by looking at the constants with the x and y terms:
  • If the equation is in the form



In our example equation
our circle is centered at the origin, with coordinates (0,0). When plotting, you start by marking this central point on your graph. This point serves as the focal reference from which you measure the radius outward in all directions to draw the full circle.

The center is crucial for constructing the graph accurately and ensuring the circle is correctly positioned.

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