Problem 45 - Precalculus Essentials

Question

Write the standard form of the equation of the circle with the given center and radius. $$(x+3)^{2}+(y-2)^{2}=4$$

Step-by-Step Solution

Answer

The equation of the circle is already in the standard form with the center at (-3, 2) and radius 2.

Step 1: Identify the Standard Form

The standard form of a circle equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Step 2: Read Off the Values

\((x + 3)^2 + (y - 2)^2 = 4\) can be written as \((x - (-3))^2 + (y - 2)^2 = 2^2\).
Center: \((-3, 2)\), Radius: \(2\).

Step 3: Conclusion

The equation is already in standard form with center \((-3, 2)\) and radius 2.

Key Concepts

Circle Equations

Understanding the equation of a circle is essential when studying geometry in precalculus. A circle's equation in the standard form is expressed as \( (x-h)^2 + (y-k)^2 = r^2 \), where \( (h,k) \) is the center of the circle, and \( r \) is its radius. This form allows us to easily identify the specific location and size of the circle on the Cartesian plane.

Identifying Circle Parameters

When given the equation \( (x+3)^2 + (y-2)^2 = 4 \), you'll notice it matches the standard form. Here, the center is found by interpreting the sign in front of the numbers inside the parentheses. The center is \( (-3,2) \) because the equation has \( x+3 \) and \( y-2 \) components, which correspond to \( x \) and \( y \) values shifted by -3 and +2 from the origin, respectively. Additionally, the radius is the square root of the number on the other side of the equation, which in this case is \( \sqrt{4} = 2 \). This concise representation makes plotting the circle straightforward, aiding in the visualization of geometric properties.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. These shapes include circles, ellipses, parabolas, and hyperbolas, each with distinct equations and properties that are studied in precalculus. Circle equations are a special case of the conic sections, where the plane cuts the cone parallel to its base.

Distinguishing Conic Sections

One way to identify a conic section equation is by its general second-degree polynomial form \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). The values and relationships between \( A \) and \( C \) determine the type of conic. For a circle, specifically, \( A \) and \( C \) are equal and the \( Bxy \) term is absent because the axes are perpendicular to each other and there is no xy term needed.

Understanding conic sections allows students to classify and analyze geometric shapes based on their algebraic equations. This connection between algebra and geometry helps to solve complex problems involving motion, optics, and more.

Geometry in Precalculus

Geometry, which is the study of shapes, sizes, and the properties of space, is incorporated into precalculus to build a foundation for understanding more advanced mathematical concepts. In precalculus, you'll often work with geometric figures on the coordinate plane — facilitating the use of algebraic methods to solve geometric problems.

Applications of Geometry in Precalculus

• The standard form equation of a circle provides a bridge between algebraic expressions and geometric understanding.
• By analyzing the properties of shapes through their equations, one can delve deeper into the study of trigonometry, limits, and eventually calculus.
• Geometry in precalculus encompasses the visualization of functions and understanding the graphs of various equations, which is critical in subsequent mathematical fields like calculus and beyond.

As such, an appreciation for the synergy between algebra and geometry is cultivated, enabling students to harness a comprehensive set of tools to tackle mathematical challenges.