Problem 65 - College Algebra with Modeling and Visualization

Question

Find the center and radius of the circle. $$ x^{2}+(y+1)^{2}=100 $$

Step-by-Step Solution

Answer

The center of the circle is (0, -1) and the radius is 10.

Step 1: Identify the Circle Equation Format

The given equation of the circle is \(x^{2}+(y+1)^{2}=100\). This resembles the standard form of a circle equation \((x-h)^{2}+(y-k)^{2}=r^{2}\), where \(h,k\) is the center, and \(r\) is the radius.

Step 2: Determine the Center of the Circle

In the equation \((x-h)^{2}+(y-k)^{2}=r^{2}\), the terms \((x-h)^{2}\) and \((y-k)^{2}\) show the deviations from the center. Here, \(x^{2}\) implies \(h=0\) and \((y+1)^{2}\) matches \((y-(-1))^{2}\), so \(k=-1\). Therefore, the center of the circle is \((0, -1)\).

Step 3: Calculate the Radius of the Circle

The right-hand side of the equation \(x^{2}+(y+1)^{2}=100\) is \(r^{2}\). Thus, \(r^{2}=100\) implies \(r=\sqrt{100}=10\). Therefore, the radius of the circle is 10.

Key Concepts

Standard Form of a Circle

The standard form of a circle is a specific way to write the equation of a circle in the coordinate plane. It is expressed as \((x-h)^2 + (y-k)^2 = r^2\). This equation reveals important information about the circle:

• \(h\) and \(k\) are the x and y coordinates of the circle's center.
• \(r\) is the radius, which is the distance from the center to any point on the circle.

To use this form, the equation you have must match the structure exactly. For example, with the equation \(x^{2}+(y+1)^{2}=100\), it fits the standard form with the identification of \((h, k)\) as \((0, -1)\) and the radius squared as 100.
This information allows us to easily identify the geometric characteristics of the circle just by looking at the equation.

Center of a Circle

Finding the center of a circle from its equation involves spotting the values of \(h\) and \(k\) in the standard form \((x-h)^2 + (y-k)^2 = r^2\). These values define the point \((h, k)\) where the circle is centered.

In the equation \(x^{2}+(y+1)^{2}=100\), there is no term with \(x-h\), suggesting \(h = 0\) because \(x - 0 = x\). For the \((y+1)^2\) term, we identify it as \(y - (-1)\) making \(k = -1\).
So, the center of this circle is at \((0, -1)\). This means if you plot this on the graph, the circle is centered at the origin but shifted 1 unit down along the y-axis.

Radius of a Circle

The radius of a circle is a crucial aspect of its geometry, indicating how large the circle is.
To extract the radius from the standard form of a circle equation, recognize that the equation's right side, given as \(r^2\), provides \(r\) when solved by taking the square root.

Consider the equation \(x^{2}+(y+1)^{2}=100\), here \(100\) is \(r^2\). Thus, solving for \(r\) involves calculating \(\sqrt{100}\), which equals 10.
Hence, the radius is 10 units. With this value, you can understand that every point on this circle is exactly 10 units away from its center, \((0, -1)\). Knowing the radius is essential for graphing circles and understanding their size relative to other geometric entities.