Problem 14 - A Graphical Approach to Precalculus with Limits

Question

Find the center-radius form for each circle satisfying the given conditions. Center \(\left(-\frac{1}{2},-\frac{1}{4}\right) ;\) radius \(\frac{12}{5}\)

Step-by-Step Solution

Answer

The center-radius form of the circle is \((x + \frac{1}{2})^2 + (y + \frac{1}{4})^2 = \frac{144}{25}\).

Step 1: Identify the Center

The center of the circle is given as \((-\frac{1}{2}, -\frac{1}{4})\). In the center-radius form of a circle equation, these values are used as \(h\) and \(k\) in the formula \((x-h)^2 + (y-k)^2 = r^2\).

Step 2: Identify the Radius

The radius of the circle is given as \(\frac{12}{5}\). This value is used as \(r\) in the formula, so we will need to square this to find \(r^2\).

Step 3: Calculate the Radius Squared

To find \(r^2\), calculate \(\left(\frac{12}{5}\right)^2 = \frac{144}{25}\). This is used in the equation in place of \(r^2\).

Step 4: Write the Center-Radius Form

Substitute the center \(h = -\frac{1}{2}\) and \(k = -\frac{1}{4}\), along with \(r^2 = \frac{144}{25}\), into the center-radius form of a circle equation: \((x + \frac{1}{2})^2 + (y + \frac{1}{4})^2 = \frac{144}{25}\).

Step 5: Verify the Form

Check that all parts have been substituted correctly. The center-radius form is now completed as \((x + \frac{1}{2})^2 + (y + \frac{1}{4})^2 = \frac{144}{25}\).

Key Concepts

Center-Radius Form

The center-radius form is a way to express the equation of a circle neatly and clearly. This method makes it easy to identify the circle's center coordinates and its radius directly from the equation. The general template for this form is \[ (x-h)^2 + (y-k)^2 = r^2 \] where:

• \(h\) and \(k\) are the x and y coordinates of the center of the circle, respectively.
• \(r\) is the radius of the circle.

Using this form allows for quick visualization of the circle's placement on a graph. For example, in our problem, the given center is \( \left(-\frac{1}{2}, -\frac{1}{4}\right) \), meaning you simply substitute these values into the equation as \(h\) and \(k\).
So, the center-radius form of the circle becomes \[ (x + \frac{1}{2})^2 + (y + \frac{1}{4})^2 = r^2. \] This equation structure offers a direct way to understand where the circle resides in a coordinate plane.

Radius Calculation

An essential part of defining a circle is knowing its radius, which is the distance from the center point to any point on the edge of the circle. The problem provides the radius as \( \frac{12}{5} \). To use the radius in the center-radius form, we need to square it.Calculating the square of the radius involves raising the given radius to the power of 2:\[\left(\frac{12}{5}\right)^2 = \frac{12^2}{5^2} = \frac{144}{25}\]Why square the radius? The equation form requires \( r^2 \), the radius squared, to complete its structure. This allows the equation to map out the circle accurately.
The squared radius, \( \frac{144}{25} \), is then placed in the equation to make the circle's size precise.
Understanding this step ensures you can accurately use the radius in different mathematical contexts, including graphing and analytical calculations.

Geometric Concepts

Understanding geometric concepts is crucial for mastering the circle equation. At the heart of geometry is the idea of symmetry and distance. Circles are perfectly symmetrical. Every point is equidistant from the center, which is fundamental to defining a circle.The center-radius form leverages this symmetry. It shows exactly where the circle's central point is and maintains equal distance (radius) to all points on its edge. When solving problems related to circles, consider these aspects:

• Positioning: The center \((-\frac{1}{2}, -\frac{1}{4})\) is crucial. It tells us exactly where the circle sits on the coordinate plane, ensuring precise plotting.
• Size: The radius \(\frac{12}{5}\) not only defines the size but also indicates how large the circle spreads from its center. Knowing this helps in comparing and contrasting different circles.

These geometric principles help in visualizing and solving various problems involving circles. Mastering them can significantly boost comprehension of not just circle equations but many other geometric shapes and their properties.