Problem 6 - Precalculus Mathematics for Calculus

Question

The graph of the equation \((x-1)^{2}+(y-2)^{2}=9\) is a circle with center (_____),(______) and radius ______

Step-by-Step Solution

Answer

Center: (1,2); Radius: 3.

Step 1: Identify the Circle's Equation Format

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

Step 2: Match the Given Equation

Given equation is \((x-1)^2+(y-2)^2=9\).Upon comparing it with the standard form, we can deduce that \(h = 1\),\(k = 2\), and \(r^2 = 9\).

Step 3: Determine the Center Coordinates

Using the values obtained, the center of the circle is \((h, k)\) which means the center is\((1, 2)\).

Step 4: Calculate the Radius

The equation given is \(r^2 = 9\).To find the radius \(r\), take the square root of 9. Thus, \(r = 3\).

Key Concepts

Standard Form of a Circle

When we talk about the standard form of a circle's equation, we are referring to a specific mathematical expression that represents a circle in a coordinate system. The standard form is given by:\[(x - h)^2 + (y - k)^2 = r^2\]where:

• \((h, k)\) are the coordinates of the center of the circle.
• \(r\) is the radius of the circle.

This form is extremely useful because it directly shows us not only where the center of the circle is located, but also how wide the circle is, through its radius. To analyze the equation of a circle, always identify the values of the center and radius first, as they give essential information about the circle's graphical representation. For example, recognizing the equation \[(x-1)^2 + (y-2)^2 = 9\] allows us to gather all the key features of the circle it represents.

Center of a Circle

The center of a circle in the standard form equation \[(x - h)^2 + (y - k)^2 = r^2\] is given by the coordinates \((h, k)\). The center is crucial as it defines the position of the circle on a coordinate plane. In our example equation \((x-1)^2 + (y-2)^2 = 9\), we can see that

• \(h = 1\)
• \(k = 2\)

This tells us that the circle's center is located at the point \((1, 2)\) on the Cartesian plane. Keeping in mind that the center is always given by these two values extracted directly from the equation will help you quickly identify the circle's exact location and plot it accurately.

Radius of a Circle

The radius of a circle is a critical parameter that indicates the distance from the center to any point on the circle's boundary. In the standard form equation \[(x - h)^2 + (y - k)^2 = r^2\],\(r^2\) represents the square of the radius. Therefore, to find the radius \(r\), you need to take the square root of \(r^2\). For the given equation \((x-1)^2 + (y-2)^2 = 9\), we have:

• \(r^2 = 9\)

Taking the square root of 9 gives us \(r = 3\). Thus, the radius of the circle is 3 units. Understanding this concept helps determine not only the size of the circle but also aids in drawing it accurately, as the circle encompasses all points that are exactly this distance away from the center.