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Chapter 7: Problem 143

Graph each circle. Identify the center and the radius. \(x^{2}+y^{2}=9\)

Short Answer

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

Step by step solution

01

Identify the Standard Form

Recognize that the given equation of the circle is in the standard form of a circle equation: \textbf{Standard form:} oindent oindent \( (x - h)^{2} + (y - k)^{2} = r^{2} \).
02

Compare with Standard Form

Compare the given equation \( x^{2} + y^{2} = 9 \) with the standard form of a circle's equation. Notice that there are no \h and \k terms, which means the center is at the origin.
03

Determine the Center

Since the given equation is \((x - 0)^{2} + (y - 0)^{2} = 9\), identify that the center is \((h, k) = (0, 0)\).
04

Find the Radius

To find the radius, set \( r^{2} = 9 \) and solve for \( r \). The radius \( r \) is obtained by \( r = \boldsymbol{3} \), since \( r^{2} = 9 \) leads to \( r = 3 \).
05

Plot the Circle

Plot the center of the circle at \((0, 0)\) on the coordinate plane. Then, use the radius length of 3 to draw the circle around the center. Ensure that every point on the circle is exactly 3 units away from the center.

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Key Concepts

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

circle equation
When dealing with circles in a coordinate plane, understanding the circle equation is crucial. The standard form of a circle's equation is equation: $$(x - h)^2 + (y - k)^2 = r^2$$The key components of this equation are
  • \((h, k)\): the center of the circle
  • \(r\): the radius of the circle
This standard form allows you to quickly identify the circle's radius and center so that you can plot it on a graph.
radius
The radius of a circle is the distance from the center to any point on the circle. In the equation $$ (x - h)^2 + (y - k)^2 = r^2 $$, \(r^2\) represents the square of the radius.Suppose we have an equation like $$x^2 + y^2 = 9$$. By comparing this with the standard form, we see
  • \(h = 0\)
  • \(k = 0\)
  • \(r^2 = 9\)
To find the radius, solve for \(r\) by taking the square root of both sides: $$r = 3$$. Thus, the radius is 3 units.
center of a circle
The center of a circle is the point from which all points on the circle are equidistant. In the equation $$(x - h)^2 + (y - k)^2 = r^2$$, the center is given by the coordinates \((h, k)\).For example, for the equation $$x^2 + y^2 = 9$$, there are no \((h, k)\) terms, which means both \(h\) and \(k\) are zero. Therefore, the center is at \((0, 0)\). To plot this circle, place a point at \((0, 0)\) and draw the circle so that every point is exactly 3 units away from the center, creating a perfect circle shape.

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