Problem 50 - Algebra and Trigonometry

Question

\(49-54\) . Find the center and radius of the circle, and sketch its graph. $$ x^{2}+y^{2}=5 $$

Step-by-Step Solution

Answer

Center: (0,0); Radius: \(\sqrt{5}\)

Step 1: Understand the Standard Form of Circle Equation

The equation of a circle in standard form is \((x-h)^2 + (y-k)^2 = r^2\) where \((h,k)\) is the center of the circle and \(r\) is the radius.

Step 2: Identify Components from Given Equation

Given the equation is \(x^{2}+y^{2}=5\). Notice that it can be rewritten as \((x-0)^2 + (y-0)^2 = 5\).

Step 3: Determine the Center of the Circle

From the rewritten form, it is clear that the center of the circle, \((h,k)\), is \((0,0)\).

Step 4: Determine the Radius of the Circle

The given equation in the form \((x-0)^2 + (y-0)^2 = 5\) shows that \(r^2 = 5\). Thus, the radius \(r = \sqrt{5}\).

Step 5: Sketch the Graph of the Circle

To sketch the graph, plot the center at \((0,0)\) and draw a circle with radius \(\sqrt{5}\). This will be a circle centered at the origin with points lying at a distance \(\sqrt{5}\) from it.

Key Concepts

Standard Form of a Circle

The standard form of a circle equation is a crucial concept to understand when dealing with circles in the coordinate plane. This form is expressed as \((x-h)^2 + (y-k)^2 = r^2\). Here, \((h, k)\) represents the center of the circle, while \(r\) is the radius. The formula captures all the important information needed to describe a circle's position and size. Understanding how to manipulate and interpret this equation can help in finding the center and the radius quickly.
To convert any given circle equation into standard form, you rewrite it so that each variable is isolated with its respective coordinate. For instance, \(x^2 + y^2 = 5\) can be rewritten as \((x-0)^2 + (y-0)^2 = 5\), indicating that the circle is centered at the origin, \((0, 0)\).

Center of a Circle

The center of a circle in the standard form equation \((x-h)^2 + (y-k)^2 = r^2\) is given by the coordinates \((h, k)\). This point serves as the midpoint of the circle, around which every point on the circle is equidistant.
Determining the center is straightforward if the equation is in standard form. Simply identify the values of \(h\) and \(k\) from \((x-h)^2\) and \((y-k)^2\), respectively. In our example, the equation \(x^2 + y^2 = 5\) transforms to \((x-0)^2 + (y-0)^2 = 5\), making the center \((0, 0)\).
For equations not in standard form, completing the square can be used to rewrite it appropriately and find the center.

Radius of a Circle

The radius of a circle is the constant distance from the center of the circle to any point on its circumference. In the standard form equation \((x-h)^2 + (y-k)^2 = r^2\), the radius \(r\) is derived by taking the square root of \(r^2\).
In the given exercise, once we have \((x-0)^2 + (y-0)^2 = 5\), it is clear that \(r^2 = 5\). Taking the square root of 5, we find that the radius \(r\) is \(\sqrt{5}\).
The value \(\sqrt{5}\) represents the consistent length from the circle's center to its perimeter, ensuring the circle is drawn precisely around its defined midpoint.

Graphing Circles

Graphing circles accurately involves plotting the center and constructing the circle based on its radius. Once the center \((h, k)\) is identified from the standard form equation, we can place it on the coordinate plane.
Next, use the calculated radius \(r\) to determine the circle’s size by marking points that are an equal distance \(r\) from the center, in all directions. In our example, the center is \((0, 0)\) and the radius is \(\sqrt{5}\).
To graph, start at the origin and draw a circle ensuring that all points on the circumference maintain the distance \(\sqrt{5}\) from the origin. Remember to make use of quadrants to maintain symmetry and ensure an evenly shaped circle.