Problem 77 - Algebra and Trigonometry

Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Analyze and graph the equation: \(25 x^{2}+4 y^{2}=100\)

Step-by-Step Solution

Answer

The equation \( 25x^2 + 4y^2 = 100 \) represents an ellipse with semi-major axis 5 and semi-minor axis 2.

Step 1 - Identify the Conic Section

Given the equation \(25x^2 + 4y^2 = 100\), recognize that this is in the standard form of an ellipse equation \(Ax^2 + By^2 = C\). Here, \(A = 25\), \(B = 4\), and \(C = 100\).

Step 2 - Rewrite in Standard Ellipse Form

Divide the entire equation by 100 to normalize it: \[ \frac{25x^2}{100} + \frac{4y^2}{100} = \frac{100}{100} \]Simplifying this gives: \[ \frac{x^2}{4} + \frac{y^2}{25} = 1 \]

Step 3 - Determine the Ellipse Parameters

Compare \( \frac{x^2}{4} + \frac{y^2}{25} = 1 \) with the standard ellipse form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a^2 = 4 \) (so, \( a = 2 \)) and \( b^2 = 25 \) (so, \( b = 5 \)). The values of \(a\) and \(b\) represent the lengths of the semi-major and semi-minor axes, respectively.

Step 4 - Sketch the Graph

Plot the vertices of the ellipse. Since \( a = 2 \) and \( b = 5 \), the vertices along the x-axis are at \( \pm 2 \) and along the y-axis are at \( \pm 5 \). Draw the ellipse centered at the origin, with horizontal and vertical lines intersecting at these points to guide the shape of the ellipse.

Step 5 - Verify

Double-check calculations to ensure vertices are plotted correctly. Revisit the transformed equation and confirm the lengths of the semi-major and semi-minor axes.

Key Concepts

Conic Sections

Conic sections are curves obtained by slicing a cone with a plane. The four basic types are circles, ellipses, parabolas, and hyperbolas. Each type has unique properties and equations.

Ellipses are one of the conic sections. They form when the cutting plane meets the base of the cone at an oblique angle. Every ellipse has a set of properties, such as a center, vertices, and axes, that define its shape.

Standard Ellipse Form

The standard ellipse form is a simplified way to represent an ellipse mathematically. For an ellipse centered at the origin, the equation is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

Here:

• \(a^2\) and \(b^2\) are constants that determine the shape and size.
• \(a\) is the length of the semi-major axis.
• \(b\) is the length of the semi-minor axis.

In the given problem, after converting the equation \(25x^2 + 4y^2 = 100\), we find the standard form to be \(\frac{x^2}{4} + \frac{y^2}{25} = 1\).

Knowing the standard form helps in easily identifying the ellipse's properties and graphing it.

Semi-Major and Semi-Minor Axes

The semi-major and semi-minor axes are critical components of an ellipse.

• The semi-major axis is the longest radius of the ellipse, running from the center to the edge along the major direction.
• The semi-minor axis is the shorter radius, running perpendicular to the semi-major axis from the center to the edge.

In this problem, we determined from the equation \( \frac{x^2}{4} + \frac{y^2}{25} = 1 \) that:

• \(a = 2\), the length of the semi-major axis.
• \(b = 5\), the length of the semi-minor axis.

These axes help in plotting vertices and sketching the ellipse accurately.

Plotting Vertices

Plotting vertices is a crucial step in graphing an ellipse.

Vertices are specific points that lie on the ellipse. They can be found using the lengths of the semi-major and semi-minor axes.

For the given problem, after identifying that \(a = 2\) and \(b = 5\):

• The vertices along the x-axis are at \(\text{-2, 0}\) and \(2, 0\).
• The vertices along the y-axis are at \(\text{0, -5}\) and \(0, 5\).

Drawing the ellipse involves connecting these vertices smoothly, ensuring that the shape remains symmetric about its axes.

Double-checking these plotted points helps ensure the accuracy of the graph drawn.