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 general form of a circle's equation

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

Step 2: Read the given equation

The given equation is \((x-1)^2 + (y-2)^2 = 9\). This matches the standard form of a circle's equation.

Step 3: Extract the center from the equation

In the equation \((x-1)^2 + (y-2)^2 = 9\), the expressions for \((h, k)\) can be seen directly. Here, \(h = 1\) and \(k = 2\). Therefore, the center of the circle is \((1, 2)\).

Step 4: Determine the radius from the equation

The equation equates to \(r^2 = 9\), so we find the radius \(r\) by taking the square root of 9. Hence, \(r = 3\).

Key Concepts

Standard Form of a Circle

To understand the equation of a circle, it's important to start with the standard form. The standard form of a circle's equation is expressed as \((x-h)^2 + (y-k)^2 = r^2\). Here, the terms \((h, k)\) represent the center of the circle, and \(r\) is the radius. The standard form is especially useful because it clearly shows these crucial details at a glance.

• The left side of the equation, \((x-h)^2 + (y-k)^2\), indicates the transformations applied to a circle's radius to shift its center from the origin to the specific point \((h, k)\).
• The right side of the equation, \(r^2\), represents the square of the radius of the circle.

When you rewrite a circle's equation using the standard form, you can easily identify both the center and the radius. For example, the equation \((x-1)^2 + (y-2)^2 = 9\) is already expressed neatly in this form. This makes it straightforward to extract information about the circle's properties.

Center of a Circle

The center of a circle is a fixed point from which all points on the circle are equidistant. In the equation \((x-h)^2 + (y-k)^2 = r^2\), the terms \((h, k)\) define the circle's center.

• The value \(h\) represents the x-coordinate of the center.
• The value \(k\) represents the y-coordinate of the center.

To find the center from the given equation \((x-1)^2 + (y-2)^2 = 9\), look at the modifications made to \(x\) and \(y\).

Finding the Center:

Take the values inside the parentheses:

• The expression \((x-1)\) indicates a shift from \(x = 0\) to \(x = 1\).
• The expression \((y-2)\) shows a shift from \(y = 0\) to \(y = 2\).

Thus, the center of the circle is \((1, 2)\). All points on this circle are exactly the same distance from this center point, establishing the circle's uniform shape.

Radius of a Circle

The radius of a circle is the constant distance from the center of the circle to any point on its edge. In the circle's standard form equation, \((x-h)^2 + (y-k)^2 = r^2\), the radius is represented as \(r\). But in this form, it's given as \(r^2\), so you'll need to take a square root to find \(r\). For example, when looking at the equation \((x-1)^2 + (y-2)^2 = 9\), you see that \(r^2 = 9\).

• Taking the square root of both sides gives \(r = \sqrt{9}\).
• This simplifies further to \(r = 3\), indicating that every point on the edge of the circle is 3 units away from the center \((1, 2)\).

Understanding the radius is key because it defines the size of the circle. Whether you're graphing the circle or solving related problems, knowing the radius allows you to predict how large the circle is and how far it extends from its center.