Problem 60 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

The center is (0, 0) and the radius is 10.

Step 1: Rewrite the standard equation of a circle

The general equation of a circle is given by \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. Let's identify these parameters.

Step 2: Identify parameters from the equation

In the equation \(x^2 + y^2 = 100\), we can see it matches the form \((x - 0)^2 + (y - 0)^2 = 100\). This means that \((h, k) = (0, 0)\) and \(r^2 = 100\).

Step 3: Calculate the radius

From \(r^2 = 100\), take the square root of both sides to find \(r\). Thus, \(r = \sqrt{100} = 10\).

Key Concepts

Center of a Circle

The center of a circle is an important aspect when understanding or working with circle equations. In the standard form of a circle equation, \((x - h)^2 + (y - k)^2 = r^2\), the center is expressed as the point \((h, k)\).

• The components \(h\) and \(k\) determine the location of the center on the Cartesian coordinate plane.
• If the center is at the origin, like in our example, it means \(h = 0\) and \(k = 0\), so the equation simplifies to \(x^2 + y^2\).

Identifying the center helps in graphing the circle and understanding its position in the coordinate system. By knowing the coordinates of the center, you can easily determine where the circle is located or how to reposition it by changing \(h\) and \(k\). Hence, the center is used not only for plotting but also in various applications like transitioning or translating circles.

Radius of a Circle

The radius of a circle, typically denoted as \(r\), is the distance from the center of the circle to any point on its circumference. In the standard circle equation form, the radius is found from \(r^2\), the term on the right side of the equation \((x - h)^2 + (y - k)^2 = r^2\).

• If you have the equation \((x - h)^2 + (y - k)^2 = r^2\), and solve for \(r\) by finding the square root of \(r^2\), you will get the radius.
• For example, in the equation \(x^2 + y^2 = 100\), since there are no adjustments for \(h\) or \(k\), simply take the square root of 100 to find the radius, which is 10.

Understanding the radius is crucial as it determines the size of the circle. The larger the radius, the bigger the circle. The concept of radius isn't only valuable for graphing purposes but is also fundamental in calculations involving the circumference and area of the circle.

Standard Form of a Circle

The standard form of a circle's equation is one of the most straightforward yet powerful tools for analyzing circular shapes in algebra. It helps to clearly display both the center and the radius of the circle, making it easier to interpret and work with.

• In the equation \((x - h)^2 + (y - k)^2 = r^2\), the terms \(h\) and \(k\) give you the center coordinates, while \(r^2\) offers the radius squared.
• This form is derived from the Pythagorean theorem, which makes it intuitive for calculating distances.

Using the standard form allows for quick transition to graphing, and it becomes simpler to identify the essential characteristics of the circle. It is especially useful when translating circles across the plane. Different forms of the circle equation, such as the general form, can be converted into this standard form for easier calculation and visualization, demonstrating its versatility in various mathematical problems.