Problem 10 - College Algebra

Question

Graph each ellipse and locate the foci. $$ \frac{x^{2}}{4 !}+\frac{y^{2}}{\frac{25}{16}}=1 $$

Step-by-Step Solution

Answer

The ellipse is centered at (0,0), with a major axis along the x-axis of length 4 and a minor axis along the y-axis of length 5/2. The foci are at (3/4, 0) and (-3/4, 0).

Step 1: Identify the center, a and b values

Firstly, identify the center (h, k) of the ellipse which is (0,0) because there's no additional number in (x-h) or (y-k). Then, the 'a' value is the square root of the denominator under x, and 'b' is the square root of the denominator under y, which are a=2 and b=\(5/4\).

Step 2: Plot the center and draw the axes

Plot the center (0,0) on graph. Then draw the major axis (along the x axis as a=2>b=\(5/4\)) with length 2a=4 and minor axis (along the y axis) with length 2b=5/2.

Step 3: Find the foci

The distance from the center to the foci is \(c = \sqrt{a^2 - b^2}\), compute this to find c = \(\sqrt{2^2 - (5/4)^2}\) = \(\sqrt{3/16}\) = 3/4 or -3/4. The foci are located on the major axis.

Step 4: Graph the ellipse

Draw a smooth curve to connect the endpoints of the major and minor axis, forming the shape of the ellipse. Indicate the foci on the graph they are located at (3/4, 0) and (-3/4, 0).

Key Concepts

Graphing Ellipses

When graphing an ellipse, the first thing to recognize is its standard form equation which typically looks like \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \). The variables \((h, k)\) specify the center of the ellipse, and \(a\) and \(b\) represent the distances from the center to the vertices on the major and minor axes, respectively. Before graphing, identify these elements:

• Center: The center \((h, k)\) is found where terms do not add or subtract from \(x\) and \(y\) - here, it's simply \((0, 0)\).
• Vertices: Measure these from the center using 'a' along the major axis and 'b' along the minor axis.

For the given equation \( \frac{x^{2}}{4} + \frac{y^{2}}{\frac{25}{16}} = 1 \), the values \(a = 2\) and \(b = \frac{5}{4}\) indicate that the major axis is along the \(x\)-axis (since \(a > b\)). Use these measurements to sketch the ellipse by marking its vertices and drawing a smooth, symmetrical shape connecting these points.

Ellipse Foci

The foci of an ellipse are key points lying along the major axis that help define its shape. To calculate the distance from the center of the ellipse to each focus (\(c\)), use the formula \(c = \sqrt{a^2 - b^2}\).Once you have the center plotted, calculating the foci involves finding the numerical value of \(c\) and counting this distance from the center along the major axis. For the exercise's ellipse:

• Calculate: Given \(a = 2\) and \(b = \frac{5}{4}\), the calculation is \(c = \sqrt{2^2 - (5/4)^2} = \sqrt{3/16}\).
• Locate: The foci will be at positions \((\frac{3}{4}, 0)\) and \((-\frac{3}{4}, 0)\), lying on the x-axis due to the ellipse's horizontal orientation.

The foci don't lie within the passed curve but aid in ensuring the ellipse maintains its geometric balance.

Conic Sections

Ellipses are part of a fascinating group of shapes known as conic sections, arising from the intersection of a plane with a double-napped cone. Where you slice the cone determines which conic section you produce:

• Circle: A special case of the ellipse where \(a = b\).
• Ellipse: Occurs when a plane cuts through both nappes of the cone at an oblique angle, resulting in an elongated circle.
• Parabola: When the plane is parallel to the slope of the cone.
• Hyperbola: When the plane slices through both nappes, not parallel to the slant height of the cone.

These curves are ubiquitous in mathematics because of their simple geometric properties and are powerful tools for solving real-world problems such as orbit calculations, motion paths, and more.Understanding the nature of conic sections helps in many advanced mathematical topics and shows how different mathematical properties link together seamlessly.