Problem 17 - Intermediate Algebra

Question

Graph each equation. \(\frac{x^{2}}{25}+\frac{y^{2}}{4}=1\)

Step-by-Step Solution

Answer

The ellipse is centered at (0, 0) with vertices at (5,0), (-5,0), (0,2), and (0,-2). The major axis is horizontal.

Step 1: Identify the Type of Conic Section

The given equation \(\frac{x^{2}}{25} + \frac{y^{2}}{4} = 1\) is in the form of \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\), which is a standard equation of an ellipse centered at the origin. Here, \(a^{2} = 25\) and \(b^{2} = 4\).

Step 2: Determine the Lengths of the Axes

From the equation, the semi-major axis (\(a\)) is \(\sqrt{25} = 5\), and the semi-minor axis (\(b\)) is \(\sqrt{4} = 2\). The larger value, \(a = 5\), indicates that the major axis is horizontal.

Step 3: Plot the Center of the Ellipse

The center of the ellipse is at the origin, \((0, 0)\), since there are no shifts in the equation.

Step 4: Plot the Vertices on the Axes

For an ellipse centered at \((0,0)\) with a major axis length of 10 (2\(a\)) along the x-axis: plot the vertices at \((5,0)\) and \((-5,0)\). For the minor axis with length 4 (2\(b\)), plot the vertices at \((0,2)\) and \((0,-2)\).

Step 5: Sketch the Ellipse

Draw a smooth curve to connect the plotted vertices, forming the shape of the ellipse. Ensure that the curve is symmetric about both axes.

Key Concepts

Ellipse

An ellipse is a fascinating shape in the realm of conic sections. It resembles an elongated circle or an oval. It is defined by two axes: the major axis and the minor axis. The major axis is the longest diameter across the ellipse, while the minor axis is the shortest diameter. These axes intersect at the center of the ellipse.

The shape of an ellipse can vary widely, from being almost circular to being quite stretched out. The key characteristic of an ellipse is that the sum of the distances from any point on the ellipse to two fixed points (called foci) is constant.

• Major axis: The longer line, dividing the ellipse into two equal halves
• Minor axis: The shorter line, perpendicular to the major axis
• Foci: Two special points inside the ellipse that help define its shape

Understanding the properties of an ellipse is essential for graphing and solving equations related to this conic section.

Graphing Conic Sections

Graphing conic sections can seem tricky, but with basic knowledge, it's quite manageable. Conic sections include ellipses, circles, parabolas, and hyperbolas. Each conic section has its own standard form equation.

To graph an ellipse, you start by identifying the center of the ellipse. This is the point where the major and minor axes intersect. Once you have the center, you can determine the lengths of the axes from the equation, as these will guide where you plot key points.

• Identify the type of conic section by its equation
• Determine the center and axes lengths (both major and minor)
• Plot the center and vertices on a graph
• Sketch the conic section, ensuring symmetry about the axes

The process emphasizes understanding the structure and properties of each shape, enabling you to represent them accurately on the graph.

Standard Equation of an Ellipse

The standard equation of an ellipse provides a simple method to identify its characteristics and graph it. This equation is typically written as \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] where \(a\) is the length of the semi-major axis and \(b\) is the length of the semi-minor axis.

The values of \(a^2\) and \(b^2\) are derived directly from this equation. If \(a > b\), the ellipse extends further along the x-axis (horizontally). Conversely, if \(b > a\), the ellipse will extend more along the y-axis (vertically).

• \(a^2\) and \(b^2\) denote the squared lengths of semi-major and semi-minor axes
• The sum of fractions equals 1 signifies the unique equation of the ellipse
• Ellipses centered at the origin have no horizontal or vertical shift

This form allows for easy graphing and understanding of how changes in \(a\) and \(b\) affect the shape and orientation of the ellipse.