Problem 13 - A Graphical Approach to Precalculus with Limits

Question

Find the center-radius form for each circle satisfying the given conditions. Center \(\left(\frac{2}{3},-\frac{4}{5}\right) ;\) radius \(\frac{3}{7}\)

Step-by-Step Solution

Answer

\((x - \frac{2}{3})^2 + (y + \frac{4}{5})^2 = \frac{9}{49}\)

Step 1: Understand the Standard Circle Equation

The standard equation of a circle in center-radius form is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) represents the center of the circle and \(r\) represents the radius. Given these values, our goal is to write the equation using the provided center and radius.

Step 2: Substitute the Center Values

Substitute \(h\) and \(k\) with their respective values from the given center \(\left(\frac{2}{3}, -\frac{4}{5}\right)\) into the circle equation. This gives us: \((x - \frac{2}{3})^2 + (y + \frac{4}{5})^2 = r^2\).

Step 3: Substitute the Radius Value

Substitute \(r\) with the given radius \(\frac{3}{7}\) and simplify \(r^2\) to complete the equation:\[ r^2 = \left(\frac{3}{7}\right)^2 = \frac{9}{49}.\]

Step 4: Write the Final Equation

Combine the substituted values into the full form of the circle's equation:\[(x - \frac{2}{3})^2 + (y + \frac{4}{5})^2 = \frac{9}{49}. \]This is the center-radius form of the circle with the given center and radius.

Key Concepts

Center-Radius Form

The Center-Radius Form of a circle equation is a straightforward way to express the equation of a circle by highlighting its geometric center and radius. Understanding this form is key to finding or verifying a circle's properties, such as its specific location in a coordinate plane and its overall size.

The Center-Radius Form is written as \((x - h)^2 + (y - k)^2 = r^2\), where:

• \((h, k)\) are the coordinates of the circle's center.
• \(r\) represents the radius.
• The equation essentially describes all the points \((x, y)\) that are \(r\) units away from the center point \((h, k)\).

When you know the circle's center as well as its radius, expressing the circle in this form helps quickly plot or visualize it as well as identify important properties such as interval ranges for x and y.

Standard Form of Circle

The Standard Form of a Circle is a mathematical equation that simplifies understanding the position and size of a circle on a coordinate plane. It’s an expansion upon the center-radius equation for educational purposes.

To recall, the center-radius form \[ (x - h)^2 + (y - k)^2 = r^2 \] can be expanded to show it in a general Standard Form \(x^2 + y^2 + Dx + Ey + F = 0\).

• Here, \(D\), \(E\), and \(F\) are constants that can be derived from the expanded form of the original circle equation.
• This form is useful in mathematical problem-solving when integrating circles into larger geometrical or analytical contexts.

Although initially more complex than the center-radius form, it offers an alternative route to solving or graphing a circle in more advanced mathematics.

Circle Geometry

Circle Geometry involves understanding the various elements and points that define a circle. Circles are unique shapes, and not only their equations but their properties and relationships are vital in different fields of mathematics. By studying these, students can solve a variety of geometric problems.

Key elements of circle geometry include:

• **Center and Radius**: As previously discussed, the center is a fixed point, and the radius is a line segment from the center to any point on the circle showing the circle's size.
• **Diameter**: The diameter is twice the radius and runs through the center, providing the widest span of the circle.
• **Circumference**: The distance around the circle, which can be calculated as \(2 \pi r\).
• **Arc**: A part of the circle's circumference, often studied in terms of angles and sections.

Understanding these structures aids not only in solving circle problems but also in applying these concepts in real-world applications like design, engineering, and physics.

Mathematical Substitution

Mathematical Substitution is a critical tool in algebra that allows for replacing variables with known values or expressions to simplify equations or solve for unknowns. This technique is particularly useful in expressing equations of circles and updating them with specific values for practical usage.

When dealing with circles, substitution is used to insert the known values of the circle's center and radius into its equation. For example:

• Start with the generic circle equation: \((x - h)^2 + (y - k)^2 = r^2\).
• Substitute the center \((h, k)\) with the specific coordinates from the problem: replace \(h\) with \(\frac{2}{3}\) and \(k\) with \(-\frac{4}{5}\).
• Insert the radius \(r\) value, for instance, \(\frac{3}{7}\), and compute \(r^2\) as \(\frac{9}{49}\).
• The updated equation then provides a clear representation of the circle's position and size.

Substitution is a flexible process key to solving many types of mathematical problems and transferring theoretical concepts into practical equations.