Problem 27 - Intermediate Algebra

Question

Graph each equation. \(\frac{(x+2)^{2}}{64}+\frac{(y-2)^{2}}{100}=1\)

Step-by-Step Solution

Answer

Plot the ellipse centered at \((-2, 2)\) with vertices at \((-2, 12)\) and \((-2, -8)\), and co-vertices at \((6, 2)\) and \((-10, 2)\).

Step 1: Identify the Type of Conic Section

Notice the equation \( \frac{(x+2)^{2}}{64}+\frac{(y-2)^{2}}{100}=1 \) is in the form \( \frac{(x-h)^{2}}{a^2}+\frac{(y-k)^{2}}{b^2}=1 \), which is the standard form of an ellipse. This means we're working with an ellipse.

Step 2: Identify the Center of the Ellipse

From the equation \( \frac{(x+2)^{2}}{64}+\frac{(y-2)^{2}}{100}=1 \), we can identify \( h = -2 \) and \( k = 2 \). Therefore, the center of the ellipse is \((-2, 2)\).

Step 3: Determine the Semi-major and Semi-minor Axes

The denominator under \( (y-2)^{2} \) is 100, which is larger than the denominator under \( (x+2)^{2} \), which is 64. This indicates that the semi-major axis is along the \( y \)-axis. Hence, \( a^2 = 100 \) and \( b^2 = 64 \). So, \( a = 10 \) and \( b = 8 \).

Step 4: Plot the Center

Plot the center of the ellipse at \((-2, 2)\) on the coordinate plane.

Step 5: Plot the Vertices

The vertices along the major axis (\( y \)-axis in this case) are \((-2, 2 + 10)\) and \((-2, 2 - 10)\), which are \((-2, 12)\) and \((-2, -8)\). Plot these points.

Step 6: Plot the Co-vertices

The co-vertices along the minor axis (\( x \)-axis in this case) are \((-2 + 8, 2)\) and \((-2 - 8, 2)\), which are \((6, 2)\) and \((-10, 2)\). Plot these points.

Step 7: Draw the Ellipse

Using the plotted center, vertices, and co-vertices, sketch the ellipse, ensuring it extends through these key points. The shape should be wider along the \( y \)-axis because \( a \) is greater than \( b \).

Key Concepts

Conic Sections

Conic sections are curves that are created by the intersection of a plane with a double-napped cone. Depending on the angle and position of the plane, you can get different types of conic sections. The primary types include circles, ellipses, parabolas, and hyperbolas. In this exercise, we focus on ellipses. Ellipses are particularly important in various fields such as astronomy and physics because they describe the orbits of planets and satellites. Understanding the properties of ellipses helps in making calculations related to orbits and trajectories. When it comes to their equations, ellipses are easily recognizable by their standard form:

• For horizontally oriented ellipses: \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1,\]where \( a > b \).
• For vertically oriented ellipses: \[\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1,\]where \( a > b \).

Graphing Ellipses

Graphing an ellipse involves several straightforward steps. Once you have the equation of the ellipse, you can directly extract key information from it. Here's a simplified guide on how to graph an ellipse like the one in the original exercise:

• Identify the center: From the equation \( \frac{(x+2)^{2}}{64}+\frac{(y-2)^{2}}{100}=1 \), the center \((h, k)\) is \((-2, 2)\).


• Determine the lengths of the semi-major and semi-minor axes: Identify the denominators \( a^2 = 100 \) and \( b^2 = 64 \). This tells you that the semi-major axis length is 10 (along the y-axis) and the semi-minor axis is 8 (along the x-axis).


• Plot the vertices and co-vertices: These are derived from the center by adding and subtracting the lengths of the axes. For example, vertices at \((-2, 12)\) and \((-2, -8)\), and co-vertices at \((6, 2)\) and \((-10, 2)\).

After plotting these points on the coordinate plane, sketch the ellipse, ensuring it passes through the vertices and co-vertices. This step helps translate the algebraic representation into a visual graph.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, provides a powerful framework for understanding and representing geometrical shapes using algebra. It allows you to analyze shapes using coordinates on a plane. With coordinate geometry, equations can represent points, lines, and curves, making it easier to solve various geometric problems involving distance, midpoints, and more.

• One of the pivotal roles of coordinate geometry is to describe and analyze ellipses through their standard or general equations. It allows you to identify essential parameters like center, foci, axes, and much more.


• In the case of ellipses, the coordinate plane helps visualize their orientation, such as determining whether the ellipse is stretched more along the x-axis or y-axis based on the associated equation's coefficients.


• This analytical approach gives a clear, structured method to handle complex geometric problems, providing a deep insight into the shape and its properties.

Understanding this concept is vital for anyone delving into higher-level mathematics, physics, or engineering.