Problem 26 - A Graphical Approach to College Algebra

Question

Determine the type of conic section represented by each equation, and graph it, provided a graph exists. $$\frac{x^{2}}{4}+\frac{y^{2}}{4}=1$$

Step-by-Step Solution

Answer

The equation represents a circle centered at the origin with radius 2.

Step 1: Identify the Equation Type

Recognize the general form of the given equation: \( \frac{x^2}{4} + \frac{y^2}{4} = 1 \). This equation is similar in structure to the standard form of an ellipse, \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a = b \). Since \( a = b \), the conic section is a circle.

Step 2: Determine the Parameters

In the equation \( \frac{x^2}{4} + \frac{y^2}{4} = 1 \), both denominators are equal, \( a^2 = b^2 = 4 \). Therefore, the radius is \( \sqrt{4} = 2 \).

Step 3: Simplify the Equation

Convert the equation into the more recognizable form of a circle: \( x^2 + y^2 = 4 \). This shows that the circle is centered at the origin \((0,0)\) with a radius of 2.

Step 4: Graph the Conic Section

To graph, draw a circle centered at the origin \((0,0)\) with radius 2. This means the circle will pass through the points \((2,0)\), \((-2,0)\), \((0,2)\), and \((0,-2)\) on the Cartesian plane.

Key Concepts

Circle

A circle is one of the most familiar shapes in geometry. It is defined as the set of all points in a plane that are equidistant from a fixed point called the center. This fixed distance is known as the radius. Circles are special because all radii are constant and equal, which makes them symmetric around their center. Circles can be found around us and are important in both mathematics and real life, from the orbits of planets to the design of wheels. Understanding a circle's properties can help in various areas of math and science.

• All points on a circle are the same distance from the center.
• The radius is the distance from the center to any point on the circle.
• The diameter is twice the length of the radius.
• Circles are symmetric in all directions from its center.

Being able to identify circles by their equations is essential for solving problems in geometry and algebra, as we will see more clearly when discussing standard forms of conic sections.

Standard Form

The standard form for the equation of a circle is crucial for easy computation and graphing. It is usually expressed as \[\left(x-h\right)^2 + \left(y-k\right)^2 = r^2\]where

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

This form is powerful because with just these parameters, you can easily plot the circle or work out essential points on it, like those that touch any axes. In the given problem, the equation simplifies to \[x^2 + y^2 = 4\]which immediately tells us the circle is centered at the origin \((0,0)\)and has a radius of 2. The form highlights not just the circle’s structure, but also aids in recognizing other conic sections, like ellipses, which require slight modifications to this form based on their different properties. Recognizing and using the standard form allows students to cross-verify their work and graph easily.

Graphing Conic Sections

Graphing conic sections involves accurately plotting the various shapes like circles, ellipses, parabolas, and hyperbolas based on their distinct equations. For circles, specifically, the process is relatively straightforward given the simplicity of their standard form. Once you have identified the form and extracted key parameters like the radius and center, you draw the circle by ensuring these conditions are met. Here’s a step-by-step guidance for graphing:

• Identify the standard form: \(x^2 + y^2 = r^2\)
• Determine the center (\(h, k\)) and radius \(r\)
• Place the center on the Cartesian plane.
• Use the radius to mark off points around the center that are equidistant, ensuring symmetry.
• Connect these points to form a smooth, rounded shape—a complete circle.

In the example problem, the circle’s simple radius of 2 means you only need to reach out from the origin to \((2, 0)\), \((-2, 0)\), \((0, 2)\), and \((0, -2)\), and connect these points to visualize the circle. Accurate graphing helps in visualization and verification of the work done, making these geometric concepts tangible for students.