Problem 63 - College Algebra with Modeling and Visualization

Question

Find the center and radius of the circle. $$ (x-2)^{2}+(y+3)^{2}=9 $$

Step-by-Step Solution

Answer

The center is (2, -3) and the radius is 3.

Step 1: Identify the standard form of the circle equation

The general 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: Compare given equation to standard form

The given equation is \((x-2)^2 + (y+3)^2 = 9\). Comparing this to the standard circle equation \((x-h)^2 + (y-k)^2 = r^2\), we identify that \(h = 2\), \(k = -3\), and \(r^2 = 9\).

Step 3: Determine the center of the circle

From the equation \(x-h\) and \(y-k\), the values of \(h\) and \(k\) identify the center of the circle. Here, \(h = 2\) and \(k = -3\), so the center is \((2, -3)\).

Step 4: Calculate the radius of the circle

The equation gives \(r^2 = 9\). To find \(r\), take the square root of both sides: \(r = \sqrt{9} = 3\).

Key Concepts

Standard Form of a Circle Equation

The standard form of a circle's equation is a way to express the circle in a mathematical language. It is written as \((x-h)^2 + (y-k)^2 = r^2\). This format makes it easy to see the circle's most important features. The variables \(h\) and \(k\) represent the coordinates of the circle's center, and \(r\) represents the radius.

Think of it like this: the expression \((x-h)^2 + (y-k)^2\) tells us how far a point \((x, y)\) is from the center \((h, k)\). If this distance is exactly \(r\), then the point lies on the circle. Hence, the circle is made up of all such points that satisfy this condition.

Understanding the standard form is crucial because it gives us a clear and complete way to define any circle just by using its center and radius.

Center of a Circle

The center of a circle is a key concept, as it is the fixed point from which all points on the perimeter of the circle are equidistant. Identifying the center helps in both graphing the circle and in solving complicated geometrical problems. In the standard form equation \((x-h)^2 + (y-k)^2 = r^2\), the center of the circle is \((h, k)\).

In the given problem, our equation is \((x-2)^2 + (y+3)^2 = 9\). Here, by comparing it to the standard form, we can see that the center \((h, k)\) is \((2, -3)\). This means every point on the circle is equally distant from the point \((2, -3)\).

Remember, determining the center is as simple as identifying \(h\) and \(k\) directly from the equation.

Radius of a Circle

The radius of a circle is the distance from its center to any point on its circumference. In the context of the standard form \((x-h)^2 + (y-k)^2 = r^2\), \(r\) is the radius, while in the equation \(r^2\) is used, requiring an extra step to find \(r\).

Specifically, you find the actual radius by taking the square root of \(r^2\). For the equation \((x-2)^2 + (y+3)^2 = 9\), you have \(r^2 = 9\). To find the radius, calculate \(\sqrt{9}\), which is \(3\).

This means every point on the circle is exactly 3 units away from the center point \((2, -3)\). The radius is an essential part of understanding the size of the circle, as it directly relates to how large or small the circle is. It's crucial for anyone to be able to determine the radius accurately to fully understand any circle described by a given equation.