Problem 74 - Contemporary Precalculus

Question

Find the equation of the circle. Center (3,1)\(;\) diameter 2.

Step-by-Step Solution

Answer

Answer: (x-3)^2 + (y-1)^2 = 1

Step 1: Find the radius of the circle

Since we are given the diameter of the circle as 2, we can find the radius by dividing the diameter by 2. The radius (r) will be: r = Diameter/2 = 2/2 = 1

Step 2: Use the standard equation of a circle

Now, we will use the equation of a circle (x-a)^2 + (y-b)^2 = r^2, where (a,b) are the center coordinates and r is the radius. In this case, we have a = 3, b = 1, and r = 1.

Step 3: Plug the values into the equation

Insert the values of a, b, and r into the standard equation of the circle: (x-3)^2 + (y-1)^2 = 1^2

Step 4: Write the final equation of the circle

Now we can simply write down the final equation of the circle: (x-3)^2 + (y-1)^2 = 1 This is the equation of the given circle with center (3,1) and diameter 2.

Key Concepts

Circle Geometry

Circle geometry is all about the properties and relationships pertaining to circles. A circle is a two-dimensional shape where all points are equidistant from a central point, known as the center. This distance is called the radius. Circles are foundational in geometry because they help us understand various spatial and mathematical concepts.

In the realm of circle geometry, there are several important components you should know about, including:

• Center: This is the point from which every point on the circle is the same distance (radius).
• Radius: The fixed distance from the center of the circle to any point on its boundary.
• Diameter: A line that passes through the center of the circle and whose endpoints lie on the circle. It's twice the radius.
• Circumference: The total distance around the circle.

Understanding these key terms is essential for solving problems involving circles, such as finding their equations or determining their geometric properties.

Circle Radius

The radius of a circle is a crucial factor in defining its size and the space it occupies on a plane. It is the distance from the center of the circle to any point along its perimeter. The radius is half the length of the diameter.

To find the radius when you have the diameter, simply divide the diameter by 2. For example, if a circle's diameter is 2 units, the radius would be:\[ r = \frac{\text{Diameter}}{2} = \frac{2}{2} = 1\]

Knowing the radius allows you to perform various calculations and use the standard equation for the circle. It is also pivotal when you need to find other properties like the area or circumference of the circle. For instance, the circumference can be calculated using the formula:\[C = 2\pi r\] where \(\pi\approx 3.14159\).
The area of a circle can be calculated using:\[A = \pi r^2\] Mastering how to work with the radius is essential for delving deeper into circle geometry.

Standard Circle Equation

The standard equation of a circle provides a concise way to express the location and size of a circle on a coordinate plane. This equation takes the form:\[(x - a)^2 + (y - b)^2 = r^2\]

In this equation:

• \((a, b)\) represents the coordinates of the circle's center.
• \(r\) is the circle's radius.

For instance, in the equation \((x - 3)^2 + (y - 1)^2 = 1\), the center of the circle is at the point (3, 1) and the radius is 1. This equation tells us that any point \((x, y)\) that satisfies the equation lies on the circle.

The beauty of this formula is its simplicity and the ability to clearly convey crucial information about a circle's position and size without needing a graph. It's a powerful tool in algebra and geometry, enabling students to manipulate and understand circles within the coordinate plane.