Problem 5 - Algebra and Trigonometry

Question

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$ \frac{x^{2}}{25}+\frac{y^{2}}{64}=1 $$

Step-by-Step Solution

Answer

The ellipse is centered at the origin, with a major axis length of 16 and a minor axis length of 10. The foci are located at a distance \(\sqrt{39}\) above and below the center on the y-axis.

Step 1 - Identify the Elements of the Given Equation

From the given equation, since \(h=k=0\) (implying the center of the ellipse is at the origin), \(a^2 = 25\) (suggesting the semi-major axis length is 5 along the x-axis), and \(b^2 = 64\) (meaning the semi-minor axis length is 8 along the y-axis).

Step 2 - Compute the Distance of the Foci from the Center

Then, proceed to compute the distance of the foci from the center which is given as \(c = \sqrt{a^2 - b^2} = \sqrt{25 - 64} = \sqrt{-39}\). Since the square root of a negative number is not a real number, it seems we made a mistake. The larger one between \(a\) and \(b\) should be selected as \(a\), so actually we have \(a=8\) and \(b=5\). Let's re-calculate \(c = \sqrt{a^2 - b^2} = \sqrt{64 - 25} = \sqrt{39}\).

Step 3 - Graph the Ellipse and Locate the Foci

Finally, graph the ellipse using the determined elements from the first step, placing the center at the origin. Draw the major axis along the y-axis with length 16 (2a) and the minor axis along the x-axis with length 10 (2b). The foci of the ellipse fall along the major axis at a distance of \(c = \sqrt{39}\) from the center.

Key Concepts

Ellipse Foci Calculation

When it comes to the intriguing world of conic sections, understanding the ellipse is crucial, particularly the concept of foci. The foci (plural for focus) of an ellipse are two fixed points located along the major axis that are equidistant from the center. The sum of the distances from any point on the ellipse to each focus is constant.

To calculate the distance of each focus from the center of an ellipse, use the formula: \[\begin{equation} c = \sqrt{a^2 - b^2} \end{equation}\], where \(a\) is the length of the semi-major axis, \(b\) is the length of the semi-minor axis, and \(c\) is the distance from the center to each focus.

An essential aspect is correctly identifying \(a\) and \(b\). The semi-major axis, \(a\), is always the larger of the two, regardless of its orientation with the coordinate axes. If you mistakenly take \(b\) as the larger value, you'll end up trying to square root a negative number, as seen in the original solution. Remember, the value under the square root in this calculation must always be positive to yield real numbers for \(c\).

Conic Sections

In the broader landscape of algebra and geometry, conic sections are the curves obtained by intersecting a plane with a double napped cone. The orientation and angle of the plane determine which conic section is produced: we can get a circle, an ellipse, a parabola, or a hyperbola.

The unique properties of ellipses make them a fascinating study. Unlike circles, which are a special type of ellipse with equal axes, typical ellipses have two distinct axes called the major and minor axes. An important property to note is how the sum of distances from the foci to any point on the ellipse remains constant.

Real-World Applications

Understanding ellipses goes beyond academics; they're prevalent in real-world applications such as planetary orbits and the design of whispering galleries and oval tracks. This constant sum property is used in acoustics and in the focusing of light and sound.

Major and Minor Axes of an Ellipse

The major and minor axes are the backbone of an ellipse's shape and structure. The major axis is the longest diameter of the ellipse, and the minor axis is the shortest. They intersect at the center of the ellipse, which is also equidistant to both foci.

The lengths of these axes provide essential information for constructing and comprehending the geometry of an ellipse. Specifically, the length of the major axis dictates the spread of the curve in one direction, while the length of the minor axis dictates the spread in the perpendicular direction.

Graphing Tips

To visualize an ellipse correctly, always remember that the length of the major axis is \(2a\) and the minor axis is \(2b\), where \(a\) and \(b\) come from the formal ellipse equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). When graphing, mark these axes on your coordinate system and plot the ellipse ensuring it aligns properly with these axes. The accuracy of the foci's position, and the ellipse as a whole, will depend on your precise identification and representation of these axes.