Problem 49 - Precalculus Mathematics for Calculus

Question

(a) Use a graphing device to sketch the top half (the portion in the first and second quadrants) of the family of ellipses \(x^{2}+k y^{2}=100\) for \(k=4,10,25,\) and 50 (b) What do the members of this family of ellipses have in common? How do they differ?

Step-by-Step Solution

Answer

All ellipses are centered at origin with same area; they differ in elongation as \(k\) increases.

Step 1: Understanding the Ellipse Equation

An ellipse with the equation \(x^2 + ky^2 = 100\) needs to be analyzed. The standard form for an ellipse centered at the origin is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). For this family of ellipses, \(a^2 = 100\) and \(b^2 = \frac{100}{k}\).

Step 2: Calculate Ellipse Axes Lengths

For different values of \(k\), calculate the axes lengths. For instance, if \(k = 4\), then \(b^2 = \frac{100}{4} = 25\), producing an ellipse with axes \(a = 10\) and \(b = 5\). Repeat this for each value of \(k = 10, 25, 50\).

Step 3: Sketch the Top Half of Each Ellipse

Using the calculated axes lengths for each value of \(k\), sketch the ellipses focusing on the top halves (where \(y \geq 0\)). The axes for these ellipses are horizontal (\(x\)) and vertical (\(y\)). Plot these sketches in the first and second quadrants since \(x\) can be both positive and negative. Use a graphing device for precise plots.

Step 4: Identify Commonalities and Differences

All the ellipses are centered at the origin and symmetric about the x-axis and y-axis. They all have a constant total area value of \(100\). The major difference is their shape; as \(k\) increases, the ellipses become more elongated in the horizontal direction because their \(y\)-components shrink, leading to larger values of \(b\).

Key Concepts

Ellipse Equation

The equation of an ellipse is fundamental to understanding its properties and graphing it correctly. In the exercise, we are given the ellipse equation as \(x^2 + ky^2 = 100\). This equation is not in the standard form, so the first step is transforming it into the standard ellipse equation: \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]For this particular family, \(a^2\) is constant at 100, simplifying our equation transformation to:\[\frac{x^2}{100} + \frac{y^2}{\left(\frac{100}{k}\right)} = 1\]This conversion helps in identifying the major and minor axes of the ellipse, as it outlines the parameters that must be adjusted based on the variable \(k\). Understanding this equation is crucial for calculating the ellipse's dimensions and becomes a stepping stone for graphing it correctly.

Major and Minor Axes

The lengths of the major and minor axes of an ellipse are fundamental characteristics derived from its equation. The squared terms in the denominator of the standard ellipse equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), denote these axes, where \(a\) represents the semi-major axis, and \(b\) represents the semi-minor axis.

• A semi-major axis, \(a\), is the longest radius. For our ellipses, \(a^2 = 100\), so \(a = 10\).
• The semi-minor axis \(b\), varies with \(k\); for example, for \(k = 4\), \(b^2 = \frac{100}{4} = 25\) and \(b = 5\).
• By increasing \(k\), the value of \(b\) decreases. This change makes the ellipse appear wider rather than taller, altering its shape.

The axes’ relative lengths determine the ellipse’s orientation and appearance on a graph. Therefore, a thorough grasp of these concepts is necessary to anticipate the ellipse's properties.

Graphing Ellipses

Graphing ellipses can be greatly simplified once their equation is in the standard form, allowing us to clearly identify the semi-major and semi-minor axes. In the exercise, we are asked to graph the top halves of the ellipses in the first and second quadrants (where \(y \geq 0\)).

• First, understand the orientation: Since \(a > b\), the ellipses are oriented horizontally.
• The sketch must be drawn focusing on the positive \(y\) direction, taking only the top half of each ellipse.

Using the calculated axes lengths for each \(k\), you can input this data into a graphing device for accuracy. It ensures the curves plotted correctly reflect the equation. Expressing from negative to positive \(x\) values allows these top halves to spread equally in the first and second quadrants, completing our plotting process successfully.

First and Second Quadrants

The first and second quadrants on a graph are important when considering the symmetry of ellipses. When asked to only plot in these regions, you focus on areas where:

• The first quadrant: both \(x\) and \(y\) are positive.
• The second quadrant: \(x\) is negative, and \(y\) is positive.

This means for our ellipses; you are only interested in parts where \(y \geq 0\). The symmetry about the x-axis ensures that merely focusing on the top halves automatically satisfies this condition. This symmetry simplifies sketching and helps focus on relevant quadrants, providing a visual of how changes in \(k\) affect ellipse orientation and spread differently across these sections. Graphing these regions helps students focus on the relevant portions and facilitates their understanding of quadrant-specific ellipse behavior.