Problem 59 - Calculus for Biology and Medicine

Question

Find the center and the radius of the circle given by the equation \((x+2)^{2}+y^{2}=25\).

Step-by-Step Solution

Answer

Center: (-2, 0); Radius: 5.

Step 1: Identify the Circle Equation Form

The given circle equation is \((x+2)^2 + y^2 = 25\). This is recognized as 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: Extract the Center from the Equation

In the equation \((x+2)^2 + y^2 = 25\), compare each part with the standard form. Notice that \((x-h)\) corresponds to \((x+2)\), which implies \(h = -2\). Since \(y^2\) indicates \(y-k\) where \(k = 0\), the center \((h, k)\) is \((-2, 0)\).

Step 3: Calculate the Radius of the Circle

The equation is \((x+2)^2 + y^2 = 25\). Here, \(r^2 = 25\). To find the radius, take the square root of both sides: \(r = \sqrt{25} = 5\). Thus, the radius is 5.

Key Concepts

Equation of a Circle

The equation of a circle in its most recognizable form is the "standard form." This form is expressed as \((x-h)^2 + (y-k)^2 = r^2\). It is a compact way to represent all the points \((x, y)\) lying on the perimeter of a circle. In this format:

• \((h, k)\) are the coordinates of the circle's center.
• \(r\) represents the radius, or the distance from the center of the circle to any of its points.

The equation displays symmetry in both the x and y directions, making it simple to visualize. When given an equation like \((x+2)^2 + y^2 = 25\), it's useful to compare it with the standard form to derive the center and the radius easily.

Circle Radius

The radius of a circle is a critical component in circle equations, representing the distance from the center to any point on the circle. In the standard form \((x-h)^2 + (y-k)^2 = r^2\), \(r^2\) is the squared radius. To find the actual radius, one simply takes the square root of \(r^2\). For example, if \(r^2 = 25\), the radius \(r\) is calculated as \(r = \sqrt{25} = 5\). This tells us the circle extends 5 units from its center in all directions, defining its size. Understanding the radius is essential not only in geometry, but in various applications such as physics and engineering where circular motion and properties are fundamental.

Circle Center

The center of a circle, indicated by \((h, k)\), plays a vital role in positioning the circle in a coordinate plane. By shifting \(h\) and \(k\), the circle moves horizontally and vertically, respectively. In the equation \((x-h)^2 + (y-k)^2 = r^2\), the center is \((h, k)\).For example, in the circle equation \((x+2)^2 + y^2 = 25\), by comparing it to the standard form, we see that the terms mean \(h = -2\) and \(k = 0\), giving us the center as \((-2, 0)\). The center defines where the circle is based on the x and y axes, and is pivotal for understanding its position and orientation. Recognizing the center enables one to determine all other attributes of the circle, like symmetry and area.