Problem 20 - Algebra 2 and Trigonometry

Question

In \(20-27\) : a. Write each equation in center-radius form. b. Find the coordinates of the center. . Find the radius of the circle. $$ x^{2}+y^{2}-25=0 $$

Step-by-Step Solution

Answer

Center: (0,0); Radius: 5

Step 1: Understand the Standard Circle Equation

The standard form for the equation of a circle is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) are the coordinates of the center of the circle and \(r\) is the radius.

Step 2: Rearrange the Given Equation

Given the equation \(x^2 + y^2 - 25 = 0\), add 25 to both sides to isolate the squared terms: \(x^2 + y^2 = 25\).

Step 3: Identify Center and Radius

In the equation \(x^2 + y^2 = 25\), compare it with the standard form \((x-h)^2 + (y-k)^2 = r^2\). We find that \(h = 0\), \(k = 0\), and \(r^2 = 25\).

Step 4: Calculate the Radius

To find the radius \(r\), take the square root of 25: \(r = \sqrt{25} = 5\).

Key Concepts

Center-Radius Form

The center-radius form of a circle is an important way to represent the equation of a circle. It simplifies understanding the geometric properties of the circle, such as its center and radius. The center-radius form is derived from the standard circle equation, which is \[(x-h)^2 + (y-k)^2 = r^2\] Here,

• \((h, k)\) are the coordinates representing the center of the circle
• \(r\) is the radius, which tells us how far any point on the circle is from the center.

This form makes it easy to visually locate and graph a circle on a coordinate plane. By directly reading from the equation, you can quickly determine the center and radius.
For an equation to be in center-radius form, its structure should match this template, making it intuitive and straightforward for analysis.

Radius Calculation

Calculating the radius of a circle from its equation is simple when it is in center-radius form. The radius, denoted by \(r\), is derived by taking the square root of \(r^2\), the term on the right side of the equation.
In our example equation \(x^2 + y^2 = 25\), which equals \[(x-0)^2 + (y-0)^2 = 5^2\] you can see that \(r^2\) equals 25.

• To find \(r\), simply calculate:\[r = \sqrt{25} = 5\]
• Thus, the radius of the circle is 5 units.

Knowing \(r\) allows you to identify the circle's scale. It indicates the fixed distance from the center to any point along the circumference of the circle, providing a geometric grasp of the circle's size.

Standard Equation of a Circle

The standard equation of a circle is widely used in analytic geometry to describe the properties and placement of a circle on the Cartesian plane. It is given by the equation:\[(x-h)^2 + (y-k)^2 = r^2\]This equation provides a formulaic way to manage calculations and show:

• The center \((h, k)\) of the circle, guiding the exact position of the circle on a plane.
• The radius \(r\), crucial to understanding the circle's scale and scope.

In simpler terms, this equation signifies that every point \((x, y)\) on the circumference is exactly \(r\) units away from the center \((h, k)\). When you compare a given equation to this form, you can instantly write the circle's characteristics, making circle-based problems much more approachable and solvable.