Problem 1 - A Graphical Approach to College Algebra

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 Standard Form

The equation given is \( x^2 + y^2 = 144 \). Compare it to the standard form formulas for conic sections.

Step 2: Check for Circle Form

The standard form of a circle is \( (x - h)^2 + (y - k)^2 = r^2 \). Here, \( h = 0 \), \( k = 0 \), and \( r^2 = 144 \), so \( r = 12 \). This matches a circle with center at \((0,0)\) and radius 12.

Step 3: Confirm No Other Forms Match

Conic sections also include ellipses, parabolas, and hyperbolas, each having distinct characteristics, like alternative terms or coefficients differing from 1. This equation does not fit any other forms except the circle form.

Key Concepts

Circle

A circle is a special type of conic section characterized by its symmetrical shape. Unlike other conic sections, a circle has no eccentricity, meaning its shape is perfectly round. Circles have some important properties:

• Center: The point that is equidistant from all points on the circle.
• Radius: The distance from the center to any point on the circle. This distance remains constant.
• Circumference: The total distance around the circle, calculated using the formula \( C = 2\pi r \).
• Area: The space inside the circle, calculated with \( A = \pi r^2 \).

Understanding these characteristics will help identify circle equations, as they will always reflect the geometric simplicity that makes circles unique.

Standard Form

The standard form of a circle's equation is crucial for identifying and understanding its properties. The equation \( (x - h)^2 + (y - k)^2 = r^2 \) represents a circle, where:

• \(h\) and \(k\) are the x and y coordinates of the center of the circle. If no values for \(h\) and \(k\) are explicitly present, they are zero, indicating the center is at the origin (0,0).
• \(r\) is the radius of the circle, which can be calculated as the square root of the number on the right of the equation.

For example, for the equation \( x^2 + y^2 = 144 \), you can translate it to the standard form by recognizing it as \( (x - 0)^2 + (y - 0)^2 = 12^2 \). This reveals the circle's center at the origin with a radius of 12.
Understanding the standard form makes it easier to identify and analyze circles without necessarily graphing them.

Equation Identification

Equation identification is key when dealing with conic sections. You need to recognize the characteristics of the equation to determine which conic section it represents:

• Circle: Look for equations in the form \( (x - h)^2 + (y - k)^2 = r^2 \), with equal coefficients for \(x^2\) and \(y^2\).
• Ellipse: Similar to a circle but with different coefficients for \(x^2\) and \(y^2\), indicated by the presence of third terms.
• Parabola: Identified by having only one squared term (either \(x^2\) or \(y^2\)).
• Hyperbola: Distinct with opposing signs in the squared terms, such as \( x^2 - y^2 \).

By analyzing the structure and components of an equation like \( x^2 + y^2 = 144 \), you confirm it belongs to the circle category, making identification quick without graphing.