Problem 46 - College Algebra

Question

Give the center and radius of the circle described by the equation and graph each equation. $$ (x+1)^{2}+(y-4)^{2}=25 $$

Step-by-Step Solution

Answer

The center of the circle is (-1, 4) and the radius is 5.

Step 1: Identify the Center

The center of the circle can be found from the equation \((x+1)^{2}+(y-4)^{2}=25\). The components with x and y inside the parenthesis will identify the circle center. Here, the center is given as (-1, 4). So a=-1 and b=4.

Step 2: Identify the Radius

The radius of the circle is given by the square root of the constant term. Hence, r = √25 = 5.

Step 3: Sketch the Circle

First draw coordinate axes, then mark a point at (-1, 4) which is the circle's center. Use it as the starting point to draw the circle with radius of 5. Remembering that the radius of a circle is the same in all directions from its center. This will give you a correct placement and size of the circle.

Key Concepts

Circle Center

The circle's center is a crucial point that helps define its position on the coordinate plane. When dealing with equations of circles in the standard form, \[(x - h)^2 + (y - k)^2 = r^2,\]identifying the center is straightforward. You look for the values of \(h\) and \(k\) in the equation, but remember that the signs are opposite when pulled from the equation. For the equation given,\((x+1)^2 + (y-4)^2 = 25,\)the center \((h, k)\) can be found by observing the terms:

• The \((x+1)\) suggests that \(h = -1\).
• The \((y-4)\) shows that \(k = 4\).

Thus, the circle's center is \((-1, 4)\). This point tells us where the circle is balanced and serves as the stepping-off point to measure its radius and position it on the graph.

Circle Radius

The radius of a circle is the distance from its center to any point on its edge. It is a constant length measured in any direction from the center.In the equation form \((x - h)^2 + (y - k)^2 = r^2,\) the value \(r^2\) is the key to finding the radius. For the equation\((x+1)^2 + (y-4)^2 = 25,\)you're looking for \(r\) such that

• \(r^2 = 25\)
• Thus, \(r = \sqrt{25} = 5\)

This radius defines the circle's size and let's you know how far from the center it will spread in all directions. Remember that the radius is always a non-negative number.

Graphing Circles

Graphing a circle can be a rewarding visualization task once you've found the necessary components like the center and radius. Here is a simple guide:- **Position the Center**: Begin by marking the center point \((-1, 4)\) on your coordinate grid.- **Draw the Axes**: Make sure your coordinate axes are clearly drawn to ensure accuracy.- **Measure the Radius**: From the center point, measure out the radius of 5 units. You can do this vertically, horizontally, and diagonally to outline the circle's extent.Ensure your circle appears smoothly rounded, maintaining its radius equally in all directions. You now have a visual representation that aligns with the equation given. Graphing circles involves understanding how the radius and center interrelate, offering a complete view of their geometry. This clarity helps when checking the correctness of both the equation and its visual graph.