Problem 66 - College Algebra with Modeling and Visualization

Question

Find the center and radius of the circle. $$ (x-5)^{2}+y^{2}=19 $$

Step-by-Step Solution

Answer

Center: (5, 0); Radius: \(\sqrt{19}\).

Step 1: Identify the Standard Form of a Circle Equation

The equation of a circle in the standard form is given by \((x-h)^2 + (y-k)^2 = r^2\) where \(h, k\) is the center of the circle and \(r\) is the radius. In this form, \(h\) and \(k\) are the x and y coordinates of the circle's center, respectively.

Step 2: Compare Given Equation with Standard Form

The given equation is \(x - 5\)^2 + y^2 = 19\. Comparing this with the standard form, we find that \(h = 5\) and \(k = 0\). Thus, the center of the circle is \(h, k = (5, 0)\).

Step 3: Find the Radius of the Circle

From the standard form equation, \(r^2 = 19\). Therefore the radius \(r\) is found by taking the square root of 19, that is \( r = \sqrt{19}.\)

Step 4: State the Solution

The center of the circle is at (5, 0) and the radius of the circle is \(\sqrt{19}\).

Key Concepts

Center of a Circle

The center of a circle is a crucial point that defines its exact location in a plane. Think of it as the circle's "heart," from which every point on the boundary is equidistant.
To find the center from an equation in the standard form,

• look at the values of \(h\) and \(k\) in \((x-h)^2 + (y-k)^2 = r^2\).

In our example, the expression \((x-5)^2+y^2=19\) shows us that \(h=5\) and \(k=0\).
Thus, the center is located at the point (5, 0).
This point is key to understanding the circle's position and helping us plot it accurately on a graph.

Radius of a Circle

The radius of a circle is the distance from its center to any point on the edge, consistently the same all around.
In the standard form of the circle's equation, this is represented by \(r\) in

• \((x-h)^2 + (y-k)^2 = r^2\).

From our equation, \(r^2 = 19\), so the radius \(r\) is simply \(\sqrt{19}\).
Understanding the radius is key to determining the size of the circle.
Although \(\sqrt{19}\) is not an integer, it tells us how large the circle is and how far points on its boundary are from its center.

Standard Form of a Circle

The standard form of a circle is an algebraic way to represent all the points that make up the circle. It makes understanding and identifying the circle's key features easy.
The general format is:

• \((x-h)^2 + (y-k)^2 = r^2\)

Here, \((h, k)\) represents the coordinates of the circle's center, and \(r\) signifies the radius.
By comparing any circle equation to this form, you can quickly extract useful information about its center and size. This makes it an invaluable tool in both geometry and algebra, enabling students to transition easily between equations and visual representations.