Problem 3 - Algebra and Trigonometry

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 Standard Form

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

Step 2: Extract the Center Coordinates

Compare the given equation \((x-1)^2 + (y-2)^2 = 9\) to the standard form. From this, we identify the values of \(h\) and \(k\). Here, \(h = 1\) and \(k = 2\).

Step 3: Center of the Circle

So, the center of the circle is at \((h, k)\) = \((1, 2)\).

Step 4: Calculate the Radius

The right side of the circle equation gives \(r^2 = 9\). Thus, we find \(r\) by taking the square root: \(r = \sqrt{9}\).

Step 5: Determine the Radius Value

\(r = \sqrt{9} = 3\), so the radius of the circle is 3.

Key Concepts

Standard Form of a Circle

The equation of a circle can be conveniently expressed in its standard form. This form is:

• \((x-h)^2 + (y-k)^2 = r^2\)

In this equation,

• \((h, k)\) represents the coordinates of the center of the circle,
• \(r\) denotes the radius of the circle.

This form makes it straightforward to visually identify the geometric characteristics of the circle, namely, where its center is located and how wide it is. It's like having a formula that directly shows you these two core features of the circle's layout on a graph.
Recognizing this standard form lets you instantly match any circle equation to these key pieces of information.

Center of a Circle

The center of a circle is a crucial point that defines its position on the coordinate plane. In the standard form of a circle's equation

• \((x-h)^2 + (y-k)^2 = r^2\)

The

• \((h, k)\)

Coordinates represent the center. In simpler terms, this is the point from which every point on the circle is equidistant.
For example, when you compare \((x-1)^2 + (y-2)^2 = 9\) to the standard form, it is clear that

• \(h = 1\),
• \(k = 2\).

Thus, the center of this specific circle is at

• \((1, 2)\).

Finding the center is essential when graphing a circle or analyzing its location in a plane. It allows you to understand its placement and orientation among other figures.

Radius of a Circle

The radius of a circle is the distance from its center to any point on its circumference. It is one of the most fundamental features as it describes the size of the circle. In the equation

• \((x-h)^2 + (y-k)^2 = r^2\),

\(r\) represents this radius.
To determine the radius from an equation, you must first identify the value of \(r^2\). Taking the equation \((x-1)^2 + (y-2)^2 = 9\), we see that

• \(r^2 = 9\).

Taking the square root of both sides gives

• \(r = \sqrt{9} = 3\).

Thus, the radius is 3 units long.
The radius is not just a mere number; it is a measurement that dictates the spread of the circle. It helps in calculating other properties such as the area and circumference of the circle.
Knowing how to derive and apply the radius will significantly aid in solving more complex geometric problems involving circles.