Problem 12 - A Graphical Approach to Precalculus with Limits

Question

Find the center-radius form for each circle satisfying the given conditions. Center \((0,0) ;\) radius 5

Step-by-Step Solution

Answer

The center-radius form of the circle is \(x^2 + y^2 = 25\).

Step 1: Understand the Center-Radius Form Equation

The center-radius form of a circle's equation is written as \[(x - h)^2 + (y - k)^2 = r^2,\]where \((h,k)\) represents the coordinates of the circle's center and \(r\) is the radius of the circle.

Step 2: Substitute the Given Center into the Equation

Given that the center of the circle is \((0, 0)\), substitute these values into the standard form equation. This means \(h = 0\) and \(k = 0\), so our equation becomes: \[(x - 0)^2 + (y - 0)^2 = r^2.\]

Step 3: Substitute the Given Radius into the Equation

The radius of the circle is given as 5. Substitute this value for \(r\) in the equation:\[(x - 0)^2 + (y - 0)^2 = 5^2.\]

Step 4: Simplify the Equation

Simplify the equation by performing the operations:\((x - 0)^2 = x^2\) and \((y - 0)^2 = y^2\). Additionally, calculate the square of the radius:\[5^2 = 25.\]Therefore, the equation becomes:\[x^2 + y^2 = 25.\]

Key Concepts

Circle Equation

The circle equation in mathematics is a formula used to define all points on a plane that are at a fixed distance, called the radius, from a specified point known as the center. This can be defined by the center-radius form, which is expressed as \((x - h)^2 + (y - k)^2 = r^2\). Here, \(h\) and \(k\) represent the coordinates of the center of the circle, while \(r\) represents the radius. This form is particularly useful because it clearly identifies the position of the center and the length of the radius. By knowing these elements, one can easily graph the circle on a Cartesian plane. Whether you are starting from the center at the origin or at any other point, this equation remains applicable. It's a staple in geometry that provides a straightforward method to depict circular shapes using algebraic expressions.

Radius of a Circle

The radius of a circle is a crucial measurement. It is the distance from the center of the circle to any point on its circumference. In the equation \((x - h)^2 + (y - k)^2 = r^2\), \(r\) denotes the radius. To determine the radius of a circle when given an equation, you can take the square root of the right-hand side value. For instance, if your equation becomes \(x^2 + y^2 = 25\), the radius \(r\) can be calculated as \(\sqrt{25} = 5\). The radius not only helps to determine the size of the circle but is also essential in numerous formulas involving circles, such as the calculation of circumference and area. Understanding the radius will assist in analyzing the properties and dimensions of the circle further.

Coordinates of the Center

The coordinates of the center of a circle provide the specific location of the circle's pivotal point around which all other points (forming the circumference) are equidistant. In the center-radius form equation \((x - h)^2 + (y - k)^2 = r^2\), \((h, k)\) are the coordinates of this center. By identifying \(h\) and \(k\), you can determine where the circle is positioned on the coordinate plane. For example, if a circle's equation is \((x - 0)^2 + (y - 0)^2 = 25\), the center of the circle is at the origin, expressed as the point \((0, 0)\). Recognizing the center is crucial not only for graphic positioning but also for calculating other specific geometric transformations or translations of the circle in coordinate geometry.