Problem 1 - A Graphical Approach to Precalculus with Limits

Question

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$x^{2}+y^{2}=144$$

Step-by-Step Solution

Answer

The equation represents a circle.

Step 1: Identify the Form

The given equation is \( x^2 + y^2 = 144 \). Compare this with standard conic section equations. In this case, the equation resembles the form \( x^2 + y^2 = r^2 \).

Step 2: Recognize the Type of Conic

The equation \( x^2 + y^2 = r^2 \) is the standard form of a circle equation. Here, both \(x\) and \(y\) variables are squared and added together, and equal a constant.

Step 3: Conclude the Graph Type

Since the equation is in the form \( x^2 + y^2 = r^2 \), it represents a circle with no coefficients discriminating between \(x\) and \(y\), confirming that it is indeed a circle.

Key Concepts

Equation of a Circle

The equation of a circle is a type of conic section that can be easily identified when the equation is in standard form. A circle's equation usually looks like this: \( x^2 + y^2 = r^2 \). Here, \(r\) is the radius of the circle, and it is squared. This standard form makes it straightforward to understand and identify the circle just by looking at the equation. When the equation is \( x^2 + y^2 = 144 \), you can immediately identify it as a circle because it matches the form \( x^2 + y^2 = r^2 \), where \(r^2 = 144\). Taking the square root of 144 gives you \(r = 12\), which is the radius of the circle. Recognizing this form is key to solving problems efficiently without graphing.

Graphing Equations

Graphing equations is a fundamental skill in understanding and visualizing how different forms relate to specific shapes. While you don’t actually have to graph the equation to recognize it as a circle, knowing how the graph of \( x^2 + y^2 = r^2 \) appears can be helpful.

• The graph of a circle is a set of points that are all equidistant from a fixed center point.
• In the standard form \( x^2 + y^2 = r^2 \), the center is at the origin, which is (0,0), given that there are no additional terms.
• The radius is equal to \(r\), which for the equation \( x^2 + y^2 = 144 \), is 12.

Understanding how the equation maps onto a graph is fundamental, even when you don't draw it. Recognizing the graph's character helps in understanding both the geometric and algebraic properties of the equation.

Standard Form

The concept of standard form is crucial when dealing with conic sections. Standard forms allow us to classify and understand equations at a glance. For a circle, as discussed, the standard form is \( x^2 + y^2 = r^2 \). This is a very simplified version, which doesn’t require you to manipulate coefficients or constants.
Compared to other conic sections:

• An ellipse, which includes different coefficients for \(x\) and \(y\), making it look like \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).
• A parabola, where you might see \( y = ax^2 \) or \( x = ay^2 \).
• And a hyperbola, which might look like \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \).

Understanding these standard forms helps not only in identifying the circle but also in distinguishing it from other conic sections. It’s a framework that makes the process of solving and graphing more systematic and less daunting.