Problem 31 - Algebra and Trigonometry

Question

Use a graphing device to graph the ellipse. $$ 6 x^{2}+y^{2}=36 $$

Step-by-Step Solution

Answer

The ellipse center is at the origin (0,0), with vertical major axis length 12 and horizontal minor axis length approximately 4.9, creating a vertically stretched ellipse.

Step 1: Write the Equation in Standard Form

Start by expressing the equation in the standard form of an ellipse. Given equation is \(6x^2 + y^2 = 36\). Divide every term by 36 to simplify:\[ \frac{6x^2}{36} + \frac{y^2}{36} = \frac{36}{36} \]This simplifies to:\[ \frac{x^2}{6} + \frac{y^2}{36} = 1 \]This is now in the standard form \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\) with \(h = 0\), \(k = 0\), \(a^2 = 6\), and \(b^2 = 36\).

Step 2: Determine Center and Axes

From the standard form of the ellipse \(\frac{x^2}{6} + \frac{y^2}{36} = 1\), identify the center \((h, k)\): - Center is at the origin \((0, 0)\).- Major axis is vertical since \(b^2 > a^2\) and \(b = \sqrt{36} = 6\).- Minor axis is horizontal with \(a = \sqrt{6} \approx 2.45\).

Step 3: Plot the Ellipse

Plot the ellipse on a graph using the center and the lengths of the axes: - From the center, go 6 units up and down for the vertical major axis. - From the center, go approximately 2.45 units left and right for the horizontal minor axis. - Draw an oval that smoothly connects these points.

Key Concepts

Graphing Ellipses

An ellipse is a beautifully symmetric shape, more stretched in one direction than a circle. When graphing an ellipse, imagine it as a stretched circle, where the distance from the center to the edge varies along different directions.
To begin, it is crucial to find the center and determine the lengths of the axes. The axes tell us how far the ellipse extends horizontally and vertically. Once these are identified, we can plot these distances from the center on a coordinate plane.
Here's a quick guide:

• Place the center of the ellipse on the graph first, it acts as your starting point.
• Mark the endpoints of the major and minor axes based on their lengths derived from the equation.
• Connect these endpoints with a smooth, oval line to form the ellipse.

By following these steps, you will be able to visualize the elliptical shape as intended by the equation.

Standard Form of an Ellipse

The standard form of an ellipse equation makes understanding the shape's properties easier. It is neatly written as \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \).
Where:

• \( (h, k) \) is the center of the ellipse.
• \( a^2 \) and \( b^2 \) represent the squares of the semi-major and semi-minor axes.

This standard form helps in quickly identifying how the ellipse is oriented and positioned. In converting an equation to this form, like \( 6x^2 + y^2 = 36 \), you make the calculation more tractable. By dividing by 36, everything fits into the standard template, exposing the ellipse's dimensions and orientation.
Such manipulations allow for a clearer sketching process, directly indicating the length of each axis and its relative direction.

Ellipse Axes

The axes of an ellipse play a pivotal role in defining its overall look.
There are two primary axes:

• The **major axis**, which is the longest diameter of the ellipse.
• The **minor axis**, which is the shortest diameter.

In the equation \( \frac{x^2}{6} + \frac{y^2}{36} = 1 \), since \( b^2 > a^2 \), the major axis is vertical, stretching 6 units above and below the center. The minor axis, meanwhile, is horizontal, stretching approximately 2.45 units to each side of the center.
Understanding these axes' directions and lengths is fundamental not just for plotting but also for interpreting the ellipse's shape in real-world applications, such as in the orbits of planets or in design architecture. The axes dictate the degree and direction of the ellipse's elongation, offering crucial insights into its geometric and practical implications.