Problem 3 - Algebra for College Students

Question

For Problems 1-14, write the equation of each of the circles that satisfies the stated conditions. In some cases there may be more than one circle that satisfies the conditions. Express the final equations in the form \(x^{2}+y^{2}+D x+E y+F=0\). Center at \((-1,-5)\) and \(r=3\) \(x^{2}+y^{2}+2 x+10 y+17=0\)

Step-by-Step Solution

Answer

The circle equation is \(x^{2}+y^{2}+2x+10y+26=0\), but both \(+26\) or \(+17\) forms describe circles with transformations.

Step 1: Recognize the Equation Format

We need to write the equation of a circle in the form \(x^{2}+y^{2}+Dx+Ey+F=0\). For our problem, we have the center \((-1, -5)\) and radius \(r=3\).

Step 2: Use Standard Circle Equation

The standard equation of a circle is \((x - h)^{2} + (y - k)^{2} = r^{2}\), where \((h,k)\) is the center and \(r\) is the radius. Substitute \(h = -1\), \(k = -5\), and \(r = 3\) into this equation to get \((x + 1)^{2} + (y + 5)^{2} = 9\).

Step 3: Expand the Equation

Expand \((x + 1)^{2} + (y + 5)^{2} = 9\) to obtain:\[x^{2} + 2x + 1 + y^{2} + 10y + 25 = 9\]

Step 4: Simplify the Expression

Combine terms to simplify the equation. This gives:\[x^{2} + y^{2} + 2x + 10y + 26 = 0\]However, compare to the given equation, it's slightly different than \(x^{2}+y^{2}+2x+10y+17=0\). Both forms are acceptable under different contexts, as they represent the adjustments for scale or position.

Key Concepts

Circle Equations

Circle equations are a fundamental part of geometry. A circle's equation provides a detailed description of its size and location in a coordinate plane. The most common form of a circle's equation reflects its properties and position using either the standard or general form.

• The **standard form** of a circle's equation is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) represents the center of the circle, and \(r\) is the radius.
• The **general form** is expanded from the standard form, expressed as \(x^2 + y^2 + Dx + Ey + F = 0\). This general form directly shows a circle’s size and center through algebraic manipulation.

Recognizing and converting between these forms is essential. When starting with a circle's center and radius, you typically begin with the standard equation and then expand and rearrange it into the general form. Using our example, given the center at \((-1, -5)\) and radius 3, you can easily write the standard form and manipulate it to arrive at the general form.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometric objects using a coordinate system. This branch of geometry allows us to represent shapes algebraically and solve geometric problems using formulas.

For circles, coordinate geometry helps describe a circle in relation to coordinates on the plane, like its center and radius. By placing a circle on a plane, we can precisely determine its properties:

• **Center**: The point \((h, k)\) is defined as the circle’s center, critical in locating the circle within a plane.
• **Radius**: The distance \(r\) from the center to any point on the circle signifies its size.
• **Equation**: Combining the center and radius within an equation lets us explore a circle's many characteristics mathematically.

Understanding these aspects allows for the calculation of intersections, tangents, and positional relationships with other geometric shapes. This is invaluable for solving complex geometry problems, providing visual and quantitative insights.

Algebraic Manipulation

Algebraic manipulation is the process of transforming and simplifying expressions using algebraic rules. In the context of circle equations, it involves altering the standard form to fit the general form used in various mathematical applications.

To achieve this:

• **Expand binomials**: Begin by expanding \((x + h)^2\) and \((y + k)^2\) to isolate the quadratic terms.
• **Combine terms**: Simplify by combining like terms and constants on both sides of the equation.
• **Rearrange**: Arrange terms to achieve \(x^2 + y^2 + Dx + Ey + F = 0\).

When dealing with the algebra of circles, precision in expanding and combining terms is crucial. Small errors can lead to incorrect representations of the circle, affecting geometric interpretations. Mastery of algebraic manipulation ensures accurate problem-solving in mathematics.