Problem 20 - Intermediate Algebra

Question

Find the center and radius of each circle and graph it. $$ (x-1)^{2}+(y+4)^{2}=9 $$

Step-by-Step Solution

Answer

The center is (1, -4) and the radius is 3.

Step 1: Identify Circle Equation

Identify that the given equation \((x-1)^{2}+(y+4)^{2}=9\) represents a circle in the standard form \((x-h)^{2}+(y-k)^{2}=r^{2}\).

Step 2: Extract Center Coordinates

From the standard form, the center of the circle \((h, k)\) can be extracted from the equation as \((h, k) = (1, -4)\).

Step 3: Determine the Radius

The radius \(r\) is the square root of the right-hand side of the equation, i.e., \(r = \sqrt{9} = 3\).

Step 4: Graph the Circle

Plot the center of the circle at \((1, -4)\) on a coordinate plane. Then draw a circle with radius 3 units, ensuring it passes through all points that are 3 units away from the center.

Key Concepts

Graphing Circles

To graph a circle on a coordinate plane, one needs to understand the relationship between the circle's equation and its graphical representation. The equation \( (x-1)^{2} + (y+4)^{2} = 9 \) describes a circle centered at \( (1, -4) \) with a radius of 3 units.
Graphing involves two main steps: plotting the center and drawing the circle with the given radius.

• First, mark the center point, which is determined by the values \( h \) and \( k \) from the circle's equation, on the coordinate plane.
• Next, from the center, measure the radius distance in all directions (up, down, left, right). Connect these points smoothly to form a perfect circle.

Remember, all points on a circle are equidistant from the center. So, using a compass can help ensure the circle remains consistent, as it acts as a fixed radius tool for accuracy and precision.

Standard Form of a Circle

The standard form of a circle's equation is \( (x-h)^{2} + (y-k)^{2} = r^{2} \). This format is essential as it reveals critical details about the circle.
The term \( (x-h) \) represents the horizontal component, and \( (y-k) \) the vertical component, defining the circle's center at \( (h, k) \).
The \( r^{2} \) part of the equation represents the radius squared. Understanding how to identify these parts helps in quickly extracting the circle's key characteristics.

• Identifying the center: Given \( (h, k) \), place these values directly into the center point coordinates.
• Determining the radius: Take the square root of \( r^{2} \) to find the radius length.

Using this standard equation helps verify whether any point \( (x, y) \) of interest falls on the circle by substituting the \( x \) and \( y \) values into the equation. The circle equation will equate if the point lies on the circle.

Finding Center and Radius of a Circle

Finding the center and radius of a circle from its equation involves identifying certain elements in the equation.
Start by comparing the given equation to the standard form \( (x-h)^{2} + (y-k)^{2} = r^{2} \). In our example, \( (x-1)^{2} + (y+4)^{2} = 9 \), we identify \( h \) and \( k \) as 1 and -4, respectively.

• The center of the circle is at \( (h, k) = (1, -4) \), by examining the terms \( (x-h) \) and \( (y-k) \).
• The equation's right side holds the radius squared, \( r^{2} = 9 \). To find the radius, take the square root: \( r = \sqrt{9} = 3 \).

Understanding these fundamental parts of the circle's equation allows us to easily find the center and radius, both crucial for graphing and deeper mathematical analysis.