Problem 26 - A Graphical Approach to Precalculus with Limits

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 a radius of 2.

Step 1: Recognize the Equation Format

The given equation is \(\frac{x^{2}}{4} + \frac{y^{2}}{4} = 1\). Noticing that both \(x\) and \(y\) terms are squared and are added with each other indicates that this is a conic section equation. Specifically, a circle or an ellipse in the form \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\). Since the denominators are equal, this is specifically an equation of a circle.

Step 2: Identify the Type of Conic

Since the coefficients are both 1 and equal (as seen from \(\frac{1}{4}\) under both variables after dividing), this confirms that the equation represents a circle. A circle's equation in standard form is \(x^2 + y^2 = r^2\).

Step 3: Determine Radius and Graph

For a circle, \(x^2/a^2 + y^2/b^2 = 1\) becomes \(x^2/4 + y^2/4 = 1\), which simplifies to \(x^2 + y^2 = 4\). The equation \(x^2 + y^2 = r^2\) where \(r^2 = 4\) indicates that \(r = \sqrt{4} = 2\). This means the radius of the circle is 2. The circle is centered at the origin (0,0) with a radius of 2.

Step 4: Graph the Circle

Plot the circle on a coordinate plane with the center at the origin (0,0). Use a compass or your plotting tool to make sure the circle's radius is 2. Mark points at (2,0), (-2,0), (0,2), and (0,-2) to help guide the shape.

Key Concepts

Circle

The concept of a circle is fundamental in understanding the types of conic sections. A circle is a set of points equidistant from a fixed point known as the center. The distance from the center to any point on the circle is the radius. In coordinate geometry, a circle's standard equation centered at the origin

• is written as: \( x^2 + y^2 = r^2 \).

The equation given in the exercise, \( \frac{x^{2}}{4} + \frac{y^{2}}{4} = 1 \), represents a circle because both variables \( x \) and \( y \) are squared and the sum equals 1 after simplifying. This indicates equal denominators, further confirming it as a circle. Hence, identifying a circle involves looking for equations where the terms include equal squared variables added together with equal coefficients.

Graphing

Graphing circles is an essential skill to visualize their properties and position on a coordinate plane. To graph a circle, one must first identify the center and the radius from the equation. In the exercise, the equation simplifies to \( x^2 + y^2 = 4 \). This tells us:

• The circle is centered at (0,0), the origin.
• The radius \((r)\) is 2, calculated from \( r = \sqrt{4} \).

To construct the graph,

• start plotting the center point at the origin.
• Measure the radius in all four directions (up, down, left, right) from the center.
• these radius points are (2,0), (-2,0), (0,2), and (0,-2).
• Using these reference points helps outline the complete circle when connected smoothly.

Equation Format

The equation format gives critical insights to determine the type of conic section represented. For circles and ellipses, the general form is \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \).

• If \( a^2 = b^2 \), then the conic section is a circle.

In our equation \( \frac{x^{2}}{4} + \frac{y^{2}}{4} = 1 \), recognizing equal fractions of \( \frac{1}{4} \) under both \( x \) and \( y \) confirms that \( a = b \). The equation becomes \( x^2 + y^2 = 4 \) after multiplying through by 4, indicating both the simplified form and that both the \( x \) and \( y \) terms contribute equally to the radius squared \( r^2 \). Understanding the equation's layout allows us to quickly identify and categorize the type of conic section seen.