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Chapter 1: Problem 88

Find the center and radius of the circle and sketch its graph $$x^{2}+y^{2}=5$$

Short Answer

Expert verified
The circle's center is (0, 0) and its radius is \(\sqrt{5}\).

Step by step solution

01

Identify the Circle's Equation Type

The given equation is in the standard form of a circle: \(x^2 + y^2 = r^2\). Here, there are no \(x\) or \(y\) terms subtracted from the squared terms, indicating that the circle is centered at the origin \((0, 0)\).
02

Find the Radius of the Circle

In the standard circle equation \(x^2 + y^2 = r^2\), the right side \(r^2\) corresponds to the square of the radius of the circle. Here, it is given that \(r^2 = 5\). We find the radius \(r\) by taking the square root of 5, hence, \(r = \sqrt{5}\).
03

Write Down the Center and Radius of the Circle

Based on our analysis, the center of the circle is \((0, 0)\) and the radius is \(r = \sqrt{5}\).
04

Sketch the Graph of the Circle

To sketch the circle, draw the center at the origin \((0, 0)\). Then, from the center, draw a circle with a radius of approximately 2.24 units (since \(\sqrt{5} \approx 2.24\)). This means the circle will extend about 2.24 units in all directions from the origin.

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Key Concepts

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

Standard Form of a Circle
When you come across a circle equation like \[x^2 + y^2 = r^2\]this is known as the "standard form of a circle." It is straightforward and provides key information at a glance. Here's what this form tells you:
  • The terms \(x^2\) and \(y^2\) suggest that the equation involves a circle.
  • If there are no additional \(x\) or \(y\) values being added or subtracted (e.g., \((x-h)^2\) or \((y-k)^2\)), the circle is centered at the origin, \((0, 0)\).
  • The number on the right side, \(r^2\), is crucial as it leads to the calculation of the radius of the circle.
Knowing the standard form simplifies the task of determining the circle's center and radius.
It's an essential tool for quickly grasping and graphing circles.
Radius Calculation
The radius of a circle is a fundamental property, and its calculation is directly tied to the standard form.
In \(x^2 + y^2 = r^2\), the value \(r^2\) stands for the radius squared.
To find the actual radius, simply take the square root:
  • The given equation \(x^2 + y^2 = 5\) implies that \(r^2 = 5\).
  • Taking the square root of both sides, we find \(r = \sqrt{5}\).
Finding the radius like this is intuitive 鈥� you are simply "unsquaring" the number on the right of your equation.
Remember, the radius represents the distance from the center of the circle to any point on its edge, giving the circle its shape and size.
Circle Graphing
Graphing a circle is easier when you understand its equation and parameters.
Since our equation \(x^2 + y^2 = 5\) is in standard form, graphing it involves a few simple steps:
  • Start by plotting the center of the circle. In this case, it's at the origin, \((0, 0)\).
  • Next, use your calculated radius, \(\sqrt{5} \approx 2.24\), to determine the extent of the circle.
  • From the center, measure approximately 2.24 units in all directions 鈥� up, down, left, and right.
  • Sketch the boundary smoothly, ensuring every part is equidistant from the center.
Envisioning the circle as extending equally from its center helps maintain symmetry and accuracy.
Practice will make you faster and more precise in crafting perfect circles on a graph.

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Most popular questions from this chapter

Factor the expression completely. (This type of expression arises in calculus when using the "Product Rule.") $$\left(x^{2}+3\right)^{-1 / 3}-\frac{2}{3} x^{2}\left(x^{2}+3\right)^{-4 / 3}$$

Factor the expression by grouping terms. x^{3}+4 x^{2}+x+4

Factor the expression completely. $$49-4 y^{2}$$

Perform the indicated operations and simplify. $$(1+2 x)\left(x^{2}-3 x+1\right)$$

Perform the indicated operations and simplify. $$(2 x-5)\left(x^{2}-x+1\right)$$
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