Problem 16 - 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. $$x^{2}=25-y^{2}$$

Step-by-Step Solution

Answer

The equation represents a circle.

Step 1: Rearrange the Equation

Start by rearranging the given equation into a standard form of a conic section. The initial equation is \(x^{2}=25-y^{2}\). Add \(y^2\) to both sides to align it more closely with a familiar form: \(x^2 + y^2 = 25\).

Step 2: Identify the Standard Form

Recall the standard forms of conic sections. The equation \(x^2 + y^2 = 25\) matches the standard form of a circle, \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle.

Step 3: Determine the Type of Conic

Since the equation matches the form \(x^2 + y^2 = 25\) and both \(x^2\) and \(y^2\) terms have the same coefficients (both are 1 and both have positive coefficients), this confirms it is a circle.

Key Concepts

Circle Equation

A circle equation represents all the points in a plane that are equidistant from a single point, known as the center. The general form of a circle equation is expressed as:

• \((x - h)^2 + (y - k)^2 = r^2\)

where

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

This particular form is easy to identify because of the squared terms and the constant on the other side of the equation. When manipulating or identifying circle equations, one major aspect to focus on is ensuring both squared terms on the left side.The standard circle equation discussed here typically has no linear components (no terms involving single powers of \(x\) or \(y\)), which simplifies analysis significantly.This conciseness helps in quick identification and verification of circle properties just by looking at the equation's structure.

Standard Form

The standard form of a conic section can provide insight into the specific type of conic without requiring graphing. Each conic—ellipse, hyperbola, parabola, and circle—has a standard form that helps identify them.For circles, the standard form is particularly straightforward:

• \((x - h)^2 + (y - k)^2 = r^2\)

Here,

• \((x - h)^2\) and
• \((y - k)^2\) imply that the center is at \((h, k)\).
• The number \(r^2\) on the equation's right side indicates the radius squared.

Thus, if you recognize this pattern in an equation like \(x^2 + y^2 = 25\), you can directly identify it as a circle without graphical analysis. This direct comparison elevates the functionality of the standard form, offering a clear and concise method for identifying and working with circle equations.

Equation Rearrangement

Rearranging equations into a more familiar form can make it much easier to identify conic sections. The original exercise starts with the equation \(x^2 = 25 - y^2\). This equation must be manipulated to fit the recognizable standard form for circles. Here's how to do it:
First, add \(y^2\) to both sides. This step restructures the equation to \(x^2 + y^2 = 25\), aligning it with the standard circle form.
Rearranging is essential because many conic sections can initially appear in non-standard forms, obscuring their properties. To successfully identify the equation type, one often needs to rearrange terms or complete square processes to bring equations closer to their standard forms.Such manipulations help reveal inherent characteristics, allowing certain terms to stand out, making identification quick and precise.