Problem 6 - A Graphical Approach to College Algebra

Question

Match each equation in Column I with the appropriate description in Column II. Do not use a calculator. A. Hyperbola; center \((2,4)\) B. Ellipse; foci \((\pm 2 \sqrt{3}, 0)\) C. Hyperbola; foci \((0, \pm 2 \sqrt{5})\) D. Hyperbola; center \((-2,4)\) E. Ellipse; center \((-2,4)\) F. Center \((0,0) ;\) horizontal transverse axis G. Ellipse; foci \((0, \pm 2 \sqrt{3})\) H. Vertical major axis; center \((2,-4)\) $$\frac{(x-2)^{2}}{9}+\frac{(y+4)^{2}}{25}=1$$

Step-by-Step Solution

Answer

The description is H: Vertical major axis; center \((2, -4)\).

Step 1: Identify the Type of Equation

The given equation is \(\frac{(x-2)^{2}}{9}+\frac{(y+4)^{2}}{25}=1\). This resembles the standard form of an ellipse, which is \(\frac{(x-h)^{2}}{a^2}+\frac{(y-k)^{2}}{b^2}=1\), where \((h,k)\) is the center. Thus, this is an equation of an ellipse.

Step 2: Determine the Center of the Ellipse

The equation \(\frac{(x-2)^{2}}{9}+\frac{(y+4)^{2}}{25}=1\) has the form \(\frac{(x-h)^{2}}{a^2}+\frac{(y-k)^{2}}{b^2}=1\), where \(h = 2\) and \(k = -4\). Therefore, the center of the ellipse is at \((2, -4)\).

Step 3: Match with the Correct Description

From the center \((2, -4)\) we can identify descriptions from Column II. The description for an ellipse with this specific center is option H: 'Vertical major axis; center \((2,-4)\).' The major axis is vertical because \(b^2 = 25\) is greater than \(a^2 = 9\).

Key Concepts

Ellipse

An ellipse is a conic section that appears as an elongated circle. It is formed by slicing a cone at an angle, with the plane not parallel to the base. Ellipses have two main axes: the major axis, which is the longest diameter, and the minor axis, the shortest. Every ellipse has a center, two foci, and is symmetrical about its axes.
The standard form of an ellipse's equation is given by:

• Horizontal major axis: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where \(a > b\).
• Vertical major axis: \( \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \), where \(a > b\).

This means the ellipse is wider if \(a\) is greater and taller if \(b\) is greater. Identifying the values of \(a\) and \(b\) helps to determine the orientation of the major axis.

Hyperbola

A hyperbola is another type of conic section. It forms when a plane intersects both nappes (the two opposite ends) of a double cone. Unlike ellipses, hyperbolas have two disconnected curves often resembling mirror images. They have two foci and two vertices connected by transverse axes.
The standard forms of a hyperbola's equation are:

• Horizontal transverse axis: \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \).
• Vertical transverse axis: \( \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \).

The key feature of hyperbolas is the negative sign between the squared terms, which differentiates them from ellipses. Understanding these forms helps identify the center and the orientation of the transverse axis.

Equation Matching

Equation matching is critical for identifying conic sections and their properties. To match an equation, we look at its form and compare it with standard equations for ellipses, hyperbolas, or other conics.
Key steps include:

• Identifying the general shape (ellipse, hyperbola, etc.) from the equation's format.
• Recognizing the center by observing the \((x-h)^2\) and \((y-k)^2\) parts.
• Comparing coefficient values to understand axis directions and lengths.

Practice with matching ensures you can quickly distinguish between different conic sections and their positions. This skill is crucial for understanding more complex geometry problems.

Center of Conic Sections

The center of conic sections is a pivotal point from which properties and characteristics are defined. For ellipses and hyperbolas, the center is given by the coordinates \((h, k)\) in their standard equations..
For an ellipse or hyperbola:

• The equation \((x-h)^2/A + (y-k)^2/B = 1\) or, for hyperbolas, \((x-h)^2/A - (y-k)^2/B = 1\) shows that \((h, k)\) is the center.
• The center often marks the midpoint of the major or transverse axis.
• For circles, which are special cases of ellipses, the center is also equidistant from all points on its curve.

Understanding the center helps in visualizing the conic and determining important attributes like foci and axes. Recognizing expressions that denote the center is vital for interpreting and solving conic problems.