Problem 15 - 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. $$\frac{(x+3)^{2}}{16}+\frac{(y-2)^{2}}{16}=1$$

Step-by-Step Solution

Answer

The equation represents a circle.

Step 1: Recognize the Equation Form

The given equation is \( \frac{(x+3)^2}{16} + \frac{(y-2)^2}{16} = 1 \). This follows the standard form \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \).

Step 2: Identify the Form Type

The standard form \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \) corresponds to the equation of an ellipse, specifically when \( a = b \), it represents a circle.

Step 3: Determine the Specific Conic Type

Given that \( a^2 = 16 \) and \( b^2 = 16 \), this simplifies to \( a = b \). Therefore, this specific equation represents a circle.

Key Concepts

ellipse

An ellipse is a fascinating type of conic section resulting from slicing a cone with a plane in a specific angled manner. It is oval-shaped, much akin to a stretched circle.

Here's how you can break down its features:

• **Axes**: An ellipse has two axes of symmetry, known as the major axis and the minor axis. The major axis is longer and lies along the ellipse's widest point, while the minor axis is shorter.
• **Center**: The center of an ellipse is the point where these axes intersect.
• **Foci**: There are two focus points within an ellipse, or foci, which lie along the major axis equidistant from the center.

Ellipses have unique properties and can be represented algebraically by the equation: \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \] where \( h \) and \( k \) specify the ellipse's center,\( a \) reflects the distance from the center to the vertices on the major axis,and \( b \) represents the distance to the vertices on the minor axis.

In an ellipse, it is typically the case that \( a eq b \). However, should \( a = b \), the ellipse becomes a special type of ellipse called a circle. Understanding these properties is key to identifying and working with elliptical equations effectively.

circle

A circle is a special case of an ellipse where both axes are of equal length. It is perfectly round, and this quality makes it one of the simplest and most symmetrical shapes imaginable. Unlike a typical ellipse, a circle does not have a major and minor axis; instead, it has a single radius that remains constant.

• **Center and Radius**: A circle's equation is typically expressed in its standard form: \[ (x-h)^2 + (y-k)^2 = r^2 \] where \( h \) and \( k \) are the coordinates of the center of the circle, and \( r \) is the radius.
• **Symmetry**: Circles have an infinite number of lines of symmetry, each passing through the center, making them not only symmetrical about any point but also rotation symmetrical.

In the context of conic sections, if you encounter an equation of the form \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \) where \( a = b \), this is indicative of a circle. This simplification highlights the circle's symmetry and is a more generalized form for conic representations. Remember, in a circle, the uniform radius makes it distinct from other forms of conic sections.

standard form of conic sections

Conic sections are constructed from the intersection of a plane with a double-napped cone, leading to distinct shapes like ellipses, parabolas, hyperbolas, and circles. Conics can be conveniently expressed through their standard equations, which is essential for identifying their types.

Let's examine the standard forms:

• **Ellipse**: \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \] Here, if \( a eq b \), the conic is an ellipse.
• **Circle**: As mentioned previously, when \( a = b \), the ellipse form denotes a circle.
• **Parabola**: \( y = ax^2 + bx + c \) or \( x = ay^2 + by + c \) is the standard form of a parabola, recognized by having only one squared term.
• **Hyperbola**: \[ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \] or its alternate orientation \[ \frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1 \]

By utilizing these standard forms, mathematicians and students alike can easily determine the type of conic section presented in an equation. Recognizing these forms simply by looking at equations enables a swift classification without the need for detailed graphing, streamlining the study of geometric properties in various contexts.