Problem 60 - Calculus for Biology and Medicine

Question

Find the center and the radius of the circle given by the equation \((x+1)^{2}+(y-3)^{2}=9\).

Step-by-Step Solution

Answer

The center of the circle is \\( (-1, 3) \\\) and the radius is 3.

Step 1: Identify the Standard Form of the Circle Equation

The given equation of the circle is \( (x + 1)^{2} + (y - 3)^{2} = 9 \). This is already in the standard form for a circle \( (x - h)^{2} + (y - k)^{2} = r^{2} \), where \( (h, k) \) is the center and \( r \) is the radius.

Step 2: Compare and Extract the Center (h, k)

By comparing the given equation \( (x + 1)^{2} + (y - 3)^{2} = 9 \) with the standard form \( (x - h)^{2} + (y - k)^{2} = r^{2} \), we recognize that \( x + 1 \) equates to \( x - (-1) \), giving \( h = -1 \), and \( y - 3 \) indicates \( k = 3 \). Thus, the center of the circle is \( (-1, 3) \).

Step 3: Calculate the Radius

From the standard form \( r^{2} = 9 \), we find that the radius \( r = \sqrt{9} \), which is \( r = 3 \). Thus, the radius of the circle is 3.

Key Concepts

Standard Form of a Circle Equation

The standard form of a circle's equation helps us easily identify the essential properties of the circle, such as the center and radius. A circle's equation in standard form is expressed as

• \[(x - h)^{2} + (y - k)^{2} = r^{2}\]
• Here,
  • \((h, k)\) represents the center of the circle.
  • \(r\) is the radius.

This form allows us to "see" the circle's properties at a glance by just comparing it with a given equation in a similar structure.

For example, the given exercise equation \[(x + 1)^{2} + (y - 3)^{2} = 9\]is already in the standard form. By comparing each part, we easily identify the values

• \(h = -1\)
• \(k = 3\)
• \(r^{2} = 9\)

This knowledge sets the stage for determining the circle's center and radius.

Center of a Circle

The center of a circle is a critical point that defines its position on a coordinate plane. It is represented by the coordinates \((h, k)\) in the circle's standard equation.

From our exercise, if the equation is\[(x + 1)^{2} + (y - 3)^{2} = 9\],compare it with \[(x - h)^{2} + (y - k)^{2} = r^{2}\].You observe that:

• \((x + 1)^{2}\) implies \(h = -1\) as it is equivalent to \((x - (-1))^{2}\).
• \((y - 3)^{2}\) indicates \(k = 3\).

Therefore, the center of this circle is \((-1, 3)\).

Understanding how to identify the center of a circle from its equation enhances your ability to manipulate and graph circles, providing insights into their geometric relationships on a plane.

Radius of a Circle

The radius of a circle is a line segment that connects the circle's center to any point on its circumference. In the circle's standard form equation, the radius is represented as \(r\), and the equation provides \(r^{2}\).

Using the exercise's equation:\[(x + 1)^{2} + (y - 3)^{2} = 9\],the part that describes the radius is\(r^{2} = 9\).To find \(r\), solve for the square root of 9:

• \(r = \sqrt{9} = 3\)

The radius is 3 units.

Knowing the radius allows you to determine the circle's scale and size, making it easier to visualize and graph. It's a simple yet powerful piece of information that completes understanding any circle's geometric shape.