Problem 10 - 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. $$\frac{(x+2)^{2}}{9}+\frac{(y-4)^{2}}{16}=1$$

Step-by-Step Solution

Answer

The equation represents a vertical ellipse.

Step 1: Recognize the Conic Section Formula

The given equation \( \frac{(x+2)^{2}}{9} + \frac{(y-4)^{2}}{16} = 1 \) is in a form that matches the standard equation of an ellipse in its center form: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where \((h,k)\) are the coordinates of the center.

Step 2: Identify Parameters and Compare

Compare the given equation with the standard form. Here, \(h = -2\), \(k = 4\), \(a^2 = 9\), and \(b^2 = 16\). This confirms that \(a^2 = 9\) and \(b^2 = 16\), and the coefficients are positive, following the condition of an ellipse.

Step 3: Determine the Type of Ellipse

Since \(a^2 < b^2\) (i.e., \(9 < 16\)), the larger denominator is associated with the vertical term \((y-k)^2\). Thus, this describes a vertical ellipse, where the major axis is along the \(y\)-direction.

Key Concepts

Ellipse

An ellipse is a geometric shape that arises as a type of conic section. It can be thought of as a stretched circle. In its standard form, an ellipse is defined by its center, two axes, and the extent of its stretching along these axes. Unlike a circle, which is uniform in all directions, an ellipse has a major axis and a minor axis, which help define its shape.
An ellipse can be written in a standard equation where the terms involving the squares of the variables are divided by constants different than zero. This nature sets it apart from other conic sections, such as parabolas and hyperbolas.
Ellipses have practical applications in physics, engineering, and astronomy, such as describing the orbits of planets. Understanding this form allows students to identify ellipses in various mathematical problems.

Standard Equation

The standard equation of an ellipse reflects the symmetry and bounds of the figure. In general form, it's written as:
\[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]
In this formula,

• \(h, k\) represents the center of the ellipse.
• \((x-h)^2\) and \((y-k)^2\) translate the ellipse horizontally and vertically.
• \a^2\ and \b^2\ determine the stretching or compressing of the ellipse in the x and y directions, respectively.

When \(a^2\) is equal to \(b^2\), it reduces to a circle. This equation is key to both constructing the ellipse graph and analyzing its properties. Differentiating an equation in this form is crucial for solving related problems and identifying the graphical nature of the equation.

Center Form

The term 'Center Form' refers to how the ellipse's equation reveals its center. In the equation \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), the center \(h, k\) represents an innate symmetry of the ellipse.
The \(h\) and \(k\) values indicate the horizontal and vertical shifts from the origin, allowing the ellipse to be positioned anywhere in the coordinate plane. Determining the center gives context to the rest of the equation and helps in plotting the ellipse on a graph.
For example, in the exercise equation:

• \(h = -2\): The ellipse is translated 2 units to the left.
• \(k = 4\): The ellipse is shifted 4 units upward.

This foundational knowledge makes understanding and identifying ellipses in mathematical problems easier.

Major Axis

In an ellipse, the **major axis** is the longest diameter, spanning the widest part of the ellipse. This axis dictates the direction of greatest stretch and is pivotal in identifying the type and orientation of the ellipse.
The major axis is along the x-axis if \(a^2 > b^2\), and along the y-axis if \(a^2 < b^2\). This simple comparison determines the orientation of the ellipse. Knowing which axis is the major one aids in visualizing and graphing the ellipse.
In the example equation, since \(b^2 = 16\) and \(a^2 = 9\), it shows that \(b^2 > a^2\), thus pointing out that the major axis lies along the y-direction. Identifying this characteristic is crucial for solving conic section problems and recognizably distinguishes ellipses from other conic sections.