Problem 62 - 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

The ellipses are symmetric around the y-axis with endpoints at \(x = \pm10\). Increasing \(k\) results in a less tall ellipse.

Step 1: Understand the General Equation of an Ellipse

The equation provided is in the form \(x^2 + ky^2 = 100\), which represents an ellipse. An ellipse in standard form is represented as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). To reach this form, each equation must be divided by 100, allowing the ellipse to be visualized more clearly.

Step 2: Rewrite the Equations in Standard Ellipse Form

For each value of \(k\), rewrite the equation:- For \(k=4\), \(\frac{x^2}{100} + \frac{y^2}{25} = 1\) - For \(k=10\), \(\frac{x^2}{100} + \frac{y^2}{10} = 1\) - For \(k=25\), \(\frac{x^2}{100} + \frac{y^2}{4} = 1\) - For \(k=50\), \(\frac{x^2}{100} + \frac{y^2}{2} = 1\) This step allows us to determine the semi-major and semi-minor axes for each ellipse.

Step 3: Sketch Each Ellipse in the First and Second Quadrants

Since the task asks for the top half of the ellipses, we will only consider \(y\geq 0\). Use a graphing calculator or software to plot each ellipse using the rearranged equations:- The ellipse for \(k=4\) will be wider in 'y', resulting in a taller shape.- As \(k\) increases, the part of the ellipse above the x-axis becomes flatter (with \(k=50\) being the flattest above the x-axis).The graphs should show only the top halves of each ellipse.

Step 4: Analyze Similarities and Differences

Common features among the ellipses include:- All are positioned symmetrically about the y-axis, extending horizontally in both the first and second quadrants.- The total horizontal span (width) is constant at 20 units across, with endpoints at \(x = \pm10\).Differences:- The value of \(k\) affects the vertical span (height), with increasing \(k\) yielding a shorter vertical section in the top half.- The overall shape becomes more compressed vertically as \(k\) increases.

Key Concepts

Standard Form of an Ellipse

Understanding the standard form of an ellipse is crucial for analyzing its shape and orientation. The standard form of an ellipse is mathematically expressed as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Here, \(a^2\) and \(b^2\) are constants that determine the lengths of the semi-major and semi-minor axes respectively.
To convert an equation into this standard form, you divide the entire equation by the constant on the right side of the equation. For example, from the exercise's equation \(x^2 + ky^2 = 100\), dividing by 100 gives you an equation suitable for comparison, like \(\frac{x^2}{100} + \frac{y^2}{b^2} = 1\).
This transformation reveals critical features of the ellipse, including its orientation and dimensions based on the values of \(a\) and \(b\). The important takeaway is that this form helps identify how elongated the ellipse is and in which direction it's stretched.

Graphing Ellipses

When graphing ellipses, the standard form makes it much easier to understand how to plot them correctly. Once you have the equation in the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), you can clearly see the semi-major and semi-minor axes.
To graph an ellipse, follow these steps:

• Identify \(a\) and \(b\) from the equation. These will give you half the width and height of the ellipse.
• Locate the center of the ellipse, which is typically the origin \((0,0)\) unless otherwise shifted.
• From the center, move \(a\) units along the x-axis in both directions and \(b\) units along the y-axis.
• Draw a smooth, oval-shaped curve connecting these points.

The exercise asked to graph only the top half of the ellipses, which implies only considering the portion where \(y \geq 0\). This simplification emphasizes symmetry and the range of the y-values for different \(k\) values in the family of ellipses.

Semi-Major Axis

The semi-major axis is one of the most critical characteristics of an ellipse. It represents the longest radius of the ellipse, extending from the center to the furthest point on the perimeter.
In the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the semi-major axis is determined by either \(a\) or \(b\), depending on which is larger. If \(a > b\), then the ellipse is stretched horizontally, and \(a\) is the semi-major axis. Conversely, if \(b > a\), the ellipse is stretched vertically, making \(b\) the semi-major axis.
Understanding the semi-major axis helps in visualizing how elongated or broad an ellipse is, which is especially crucial when comparing members of a family of ellipses, like in the given exercise.

Semi-Minor Axis

The semi-minor axis is the shorter radius of the ellipse, which complements the semi-major axis and helps define the overall shape.
In the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), if \(a > b\), \(b\) becomes the semi-minor axis. If \(b > a\), then \(a\) is the semi-minor axis instead.
This concept is vital because it permits a deeper understanding of an ellipse's proportions. The semi-minor axis dictates how narrow the elliptical shape will be in comparison to its lengthier counterpart.
In the context of the exercise, as the value of \(k\) changes, so do the semi-minor axes for each ellipse, influencing how stretched or compressed they appear vertically. This variance highlights the diversity within a family of ellipses.