Problem 1 - Algebra and Trigonometry

Question

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$ \frac{x^{2}}{16}+\frac{y^{2}}{4}=1 $$

Step-by-Step Solution

Answer

The graph of the given ellipse has its foci approximately at points (3.46, 0) and (-3.46, 0). The semi-major axis is 4 and the semi-minor axis is 2.

Step 1: Identify the semi-major and semi-minor axes

Given the ellipse equation, the denominator under the \(x^{2}\) term represents \(a^{2}\) and the denominator under the \(y^{2}\) term represents \(b^{2}\), where ‘a’ is the semi-major axis and ‘b’ is the semi-minor axis. Thus, we can say that \(a^{2} = 16\) and \(b^{2} = 4\), implying that \(a = 4\) and \(b = 2\).

Step 2: Draw the Ellipse

While drawing an ellipse, the semi-major axes act as horizontal radii and semi-minor axes as vertical radii. Draw a horizontal line at the lengths of ‘a’ and ‘-a’, we get points at (4, 0) and (-4, 0). Similarly, draw a vertical line at the lengths of ‘b’ and ‘-b’, we get points at (0, 2) and (0, -2). Now, draw a smooth curve to get the ellipse.

Step 3: Determine the Foci

Foci of the ellipse lie on the major axis at a distance of \(c = \sqrt{a^{2}-b^{2}}\) from the center. Here, this gives \(c = \sqrt{16-4} = \sqrt{12}\) which is approximately equal to 3.46. Hence, the foci are approximately at the points (3.46, 0) and (-3.46, 0).

Key Concepts

Semi-Major and Semi-Minor Axes

Understanding the semi-major and semi-minor axes is essential to graphing ellipses. These axes are akin to the 'radii' of the ellipse, determining its width and height. If you approached an ellipse the way you would a circle, you would locate the longest diameter that passes through the center. This diameter is the major axis, and half of this length is called the semi-major axis (commonly denoted as ‘a’). Similarly, the shortest diameter is the minor axis, with half of that being the semi-minor axis (denoted as ‘b’).

In our example, \(a = 4\) and \(b = 2\), with the square of these values given in the equation. When you begin to sketch the ellipse, these values help you establish the furthest points on the horizontal and vertical axes from the center, setting the boundaries of the ellipse's shape. As a general rule, if \(a > b\), the ellipse is wider than it is tall, stretching along the x-axis; if \(a < b\), it stretches along the y-axis.

Ellipse Equation

The equation of an ellipse is the blueprint that helps us graph the shape on a coordinate plane. It typically takes the form \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] where \(a\) and \(b\) stand for the lengths of the semi-major and semi-minor axes, respectively. This equation represents a standardized ellipse centered at the origin \(0, 0\).

Comparing this with the given exercise, \(\frac{x^2}{16} + \frac{y^2}{4} = 1\), you can immediately identify \(a^2 = 16\) and \(b^2 = 4\) leading to their respective axis lengths. With these pieces of information, you can accurately graph the ellipse by plotting these axes. These values determine the 'shape' and 'size' of the ellipse and are pivotal in sketching a precise graph of the ellipse.

Locating Foci of an Ellipse

The foci (plural of focus) of an ellipse are two specific points on the major axis that are equidistant from the center of the ellipse. These points are crucial in the definition and construction of an ellipse. The distance of each focus from the center is denoted as 'c', and it can be found by the equation \[c = \sqrt{a^2 - b^2}\].

Why are the foci important? They hold a unique property: for any point on the ellipse, the sum of the distances to the two foci is constant and equal to the length of the major axis. This characteristic can be used for various applications, including in acoustics and orbital mechanics.

In our example, we compute the foci using the relation \(c = \sqrt{16 - 4}\) to find that \(c\) is approximately 3.46. This means the foci are located at \(\pm3.46, 0\). When graphing the ellipse, marking these foci helps to validate the correctness of the ellipse’s shape and ensures a deeper understanding of its geometric properties.