Problem 36 - A Graphical Approach to College Algebra

Question

Determine the type of conic section represented by each equation, and graph it, provided a graph exists. $$3 x^{2}+12 x+3 y^{2}=0$$

Step-by-Step Solution

Answer

The equation represents a circle with center (-2, 0) and radius 2.

Step 1: Identify the Terms

Start by examining the given equation: \(3x^2 + 12x + 3y^2 = 0\). Notice that it contains both \(x^2\) and \(y^2\) terms, and they have the same coefficients up to a multiplicative factor (3).

Step 2: Factor Out Common Factor

Factor 3 out of the entire equation to simplify it. The equation becomes: \(3(x^2 + 4x + y^2) = 0\), which simplifies to \(x^2 + 4x + y^2 = 0\) upon dividing every term by 3.

Step 3: Complete the Square

To complete the square, focus on the \(x^2 + 4x\) part. Add and subtract 4 to create a perfect square trinomial: \(x^2 + 4x = (x + 2)^2 - 4\). Thus, the equation becomes \((x + 2)^2 - 4 + y^2 = 0\) or, rearranged, \((x + 2)^2 + y^2 = 4\).

Step 4: Recognize the Conic Section

Notice that \((x + 2)^2 + y^2 = 4\) resembles the equation of a circle \((x-h)^2 + (y-k)^2 = r^2\) with center \((-2, 0)\) and radius \(2\).

Step 5: Conclusion

The given equation represents a circle since both \(x^2\) and \(y^2\) terms have the same coefficient and can be rearranged in the standard circle form.

Key Concepts

Circle

A circle is a fascinating geometric shape and an essential concept in conic sections. It is defined as the set of all points in a plane that are equidistant from a given fixed point, known as the center. The distance from the center to any point on the circle is called the radius.Understanding the equation of a circle helps in identifying it in algebraic form. The standard form of a circle's equation is:\[ (x-h)^2 + (y-k)^2 = r^2 \]where

• \((h, k)\) is the center of the circle, and
• \(r\) is the radius.

In the solved exercise, the equation given is \[ 3(x^2 + 4x + y^2) = 0\]which was transformed into the recognizable circle equation format:\[ (x + 2)^2 + y^2 = 4 \]Here, the circle has a center at \[(-2, 0)\] and a radius of 2. Understanding these transformations forms a strong foundation for identifying and graphing circles.

Completing the Square

Completing the square is a technique used to manipulate quadratic expressions so they form perfect square trinomials. This method is particularly useful when working with conic sections as it aids in rewriting equations into forms that are easier to interpret and graph.Let's take the expression:

• \(x^2 + 4x\)

To complete the square, follow these steps:- Divide the coefficient of \(x\) by 2, which gives 2.- Square this result to get 4.- Add and subtract this number inside the expression: \(x^2 + 4x + (4 - 4)\). This transforms into:\((x + 2)^2 - 4\).By completing the square, the equation becomes more evident as a circle's equation, making it easier to identify the center and radius. This technique is not only handy for circles but also other conic sections like parabolas and ellipses.

Graphing Equations

Graphing equations is a crucial skill in mathematics that translates algebraic equations into visual representations. This process helps in understanding the nature of conic sections, like circles, by observing their positions and dimensions on a graph.To graph the circle from the equation \[(x + 2)^2 + y^2 = 4\], you first identify the circle's center and radius:

• Center: \((-2, 0)\)
• Radius: 2

Next, plot the center on the coordinate plane at \((-2, 0)\). The radius signifies how far you draw the circle from its center in all directions. As the radius is 2, count two units out from the center both horizontally and vertically.Mark these points and ensure they are equidistant from the center before sketching the circle to capture its symmetrical shape.Graphing conic sections like circles not only demonstrates the practical application of algebraic manipulation but also enhances spatial reasoning and visualization skills.