Problem 14 - Beginning and Intermediate Algebra

Question

Identify the center of each ellipse and graph the equation. $$\frac{x^{2}}{36}+\frac{y^{2}}{4}=1$$

Step-by-Step Solution

Answer

The center of the ellipse is (0, 0) with major axis length 12 and minor axis length 4. To graph it, first plot the center at (0, 0). Then, draw horizontal and vertical line segments representing the major and minor axes with endpoints at (-6, 0), (6, 0), (0, -2), and (0, 2). Finally, sketch the ellipse passing through these endpoints and enclosing the center.

Step 1: Rewrite the given equation in standard form

The given equation is: $$\frac{x^{2}}{36}+\frac{y^{2}}{4}=1$$ It is already in the standard form of the ellipse, with the center (h, k) equals to (0, 0), a=6 and b=2.

Step 2: Identify the center

By comparing the equation to the standard form, we can see that the center of the ellipse is (h, k) = (0, 0).

Step 3: Determine the major and minor axes

From the given equation, we have a^2 = 36 and b^2 = 4. Therefore, we have: a = 6 (major axis) b = 2 (minor axis)

Step 4: Plot the ellipse

1. Plot the center at (0, 0). 2. Draw a horizontal line segment of length 12 along the x-axis with endpoints at (-6, 0) and (6, 0). 3. Draw a vertical line segment of length 4 along the y-axis with endpoints at (0, -2) and (0, 2). 4. Sketch the ellipse that touches these endpoints and encloses the center. The ellipse graph would have its center at the origin (0, 0) with the major axis of length 12 and minor axis of length 4.

Key Concepts

Standard Form of Ellipse

An ellipse is a set of points where the sum of the distances from two fixed points, called foci, is constant. The equation for an ellipse can be quite complex in terms of foci or directrices, but a more practical and commonly used form in mathematics is its standard form. This makes it straightforward to understand and graph.The standard form equation of an ellipse centered at (h, k) is given by:\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]- Here, \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively.- If \(a > b\), the ellipse is oriented horizontally; if \(b > a\), it is oriented vertically.- The expression is set equal to 1, ensuring all terms are simplified and proper fractions. It is also interesting to note that \(a\) is always associated with the larger denominator in the fraction whether it is under \(x\) or \(y\).
In the provided problem, the equation \( \frac{x^{2}}{36} + \frac{y^{2}}{4} = 1 \) is already in the standard form, with a horizontal orientation of the ellipse.

Center of Ellipse

The center of an ellipse is the midpoint of the major and minor axes. It is denoted by the coordinates (h, k) in the standard equation.- In most practical examples, this is an essential starting point because it anchors the entire shape of the ellipse on a coordinate plane.- When the ellipse equation is in standard form, the values of \(h\) and \(k\) are found directly in the equation, making it easy to identify.
For our example, since the equation \( \frac{x^2}{36} + \frac{y^2}{4} = 1 \) shows no shifts in \(x\) or \(y\) (meaning no \(h\) or \(k\) values exist outside of 0), the center of the ellipse is simply at the origin point, (0, 0). This ensures that the graph is centered around the main crossing point of the coordinate axes, providing symmetry and balance to the shape.

Major and Minor Axes

The major axis is the longest diameter of the ellipse, while the minor axis is the shortest. These axes help define the shape and orientation of the ellipse.- The lengths of these axes can be found by analyzing the denominators in the standard form equation. - Specifically, the term under the larger denominator relates to the major axis. In the standard form, \(a^2\) and \(b^2\) reflect the squared values of half the axes' lengths, with \(a\) being greater than or equal to \(b\).For the purposes of this exercise:- \(a^2 = 36 (a=6)\), indicating a major axis length of 12 \(since 2a = 12\).- \(b^2 = 4 (b=2)\), suggesting a minor axis length of 4 \(as 2b = 4\).- Typically, the major axis runs along the x-axis for ellipses in horizontal form while the minor axis aligns vertically. This gives the characteristic stretched circle shape with symmetry around both axes. Graphically, you would span the major axis endpoints at \((-6, 0)\) and \((6, 0)\), and minor axis endpoints at \((0, -2)\) and \((0, 2)\), then connect smoothly to form the ellipse. Both axes not only describe the size and orientation, but they are also critical to accurately plotting the ellipse on a coordinate plane.