Problem 4 - Algebra and Trigonometry

Question

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$ \frac{x^{2}}{16}+\frac{y^{2}}{49}=1 $$

Step-by-Step Solution

Answer

The graph is an ellipse centered at the origin, with semi-minor axis of length 4 along the x-axis, and semi-major axis of length 7 along the y-axis. The foci are located at (0, ±\sqrt{33}).

Step 1: Identify the Semi-Axes

Look at the coefficients under \(x^{2}\) and \(y^{2}\) in the equation. The square root of these coefficients are the lengths of the semi-major and semi-minor axes respectively. If \(b > a\), then \(b\) is the semi-major axis and \(a\) is the semi-minor axis, else vice versa. In this case we have \(a^2 = 16\) and \(b^2 = 49\), so \(a = 4\) and \(b=7\).

Step 2: Draw the Ellipse

Draw a horizontal line as the x-axis and a vertical line as the y-axis. The center of the ellipse is the origin (0, 0). Draw the ellipse by marking the points (0, ±b) and (±a, 0), then sketch the curve of the ellipse through these points.

Step 3: Find the Distance to the Foci

The foci are located at a distance of \(c = \sqrt{b^{2}-a^{2}}\) from the center along the major axis. Here, this gives: \(c=\sqrt{49-16}=\sqrt{33}\).

Step 4: Locate the Foci

Draw the foci on the major axis, ±c units away from the center along the y-axis for a vertical ellipse. Here, this is at points (0, ±\sqrt{33})

Key Concepts

Graphing Ellipses

When graphing ellipses, it's important to understand their fundamental shape and position on a coordinate plane. An ellipse is a set of all points for which the sum of the distances to two fixed points, called foci, is constant. The equation of an ellipse in the standard form is \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] where \(a\) and \(b\) represent the lengths of the semi-axes. Start by drawing the coordinate axes. The ellipse in question has its center at the origin (0, 0). Use the given axes lengths to plot the ellipse by marking points at the ends of both axes. For this specific ellipse, plot points at (0, ±7) for the vertical axis and (±4, 0) for the horizontal axis. Then trace a smooth oval through these points to complete the graph. This shape is symmetric along both axes and highlights the unique properties of ellipses.

Semi-Major Axis

The semi-major axis of an ellipse is the longest radius, stretching from the center to the perimeter along the major axis. It plays a critical role in defining the ellipse's size and orientation. In the equation \[\frac{x^2}{16} + \frac{y^2}{49} = 1\] we identify the semi-major axis by comparing the denominators. Since \(b^2 = 49\) is greater than \(a^2 = 16\), the vertical direction is the semi-major axis with length \(b = 7\). In general:

• If \(b > a\), the ellipse is vertically oriented.
• If \(a > b\), the ellipse is horizontally oriented.

This length is from the center along the y-axis, emphasizing its importance in the orientation and height of the ellipse.

Semi-Minor Axis

The semi-minor axis is the shortest radius of an ellipse, stretching from the center to the edge along the minor axis. It complements the semi-major axis, contributing to the overall ellipse shape. In the example \(\frac{x^2}{16} + \frac{y^2}{49} = 1\), the denominator \(a^2 = 16\) corresponds to the semi-minor axis, giving a length of \(a = 4\). This measurement is aligned horizontally:

• The minor axis determines how wide the ellipse will be.
• It helps create the curvature of the horizontal points.

Understanding both axes is essential for correctly scaling and graphing the ellipse.

Foci of an Ellipse

The foci of an ellipse are two special points located along the major axis that help define its shape. They are used to mathematically define the ellipse by maintaining a constant sum of distances from any point on this shape. For an ellipse centered at the origin, the distance to the foci, \(c\), can be found using:\[c = \sqrt{b^2 - a^2}\] In our scenario, \(b = 7\) and \(a = 4\), so \[c = \sqrt{49 - 16} = \sqrt{33}\].This means that the foci are located at (0, ±\(\sqrt{33}\)) along the vertical axis:

• They are positioned symmetrically between the semi-major axis.
• The presence of the foci provides the ellipse its characteristic oval shape.

Knowing the foci locations is crucial for solving more complicated problems involving ellipses.