Problem 45 - College Algebra with Modeling and Visualization

Question

Sketch a graph of the ellipse. Identify the foci and vertices. $$ \frac{(x+4)^{2}}{16}+\frac{(y-2)^{2}}{9}=1 $$

Step-by-Step Solution

Answer

The ellipse's graph is centered at \((-4, 2)\) with vertices at \((-8, 2)\) and \((0, 2)\), and foci at \((-4 - \sqrt{7}, 2)\) and \((-4 + \sqrt{7}, 2)\).

Step 1: Identify the Center

The standard form of an ellipse is \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \) where \((h, k)\) is the center of the ellipse. From the given equation \( \frac{(x+4)^{2}}{16} + \frac{(y-2)^{2}}{9} = 1 \), compare it with the standard form: \( h = -4 \) and \( k = 2 \). Thus, the center of the ellipse is \((-4, 2)\).

Step 2: Determine the Lengths of the Axes

Identify \(a^2 = 16\) and \(b^2 = 9\) from the given equation. This implies \(a = 4\) and \(b = 3\). Since \(a > b\), the major axis is horizontal. The length of the major axis is \(2a = 8\) and the length of the minor axis is \(2b = 6\).

Step 3: Plot the Vertices

The distance \(a = 4\) is along the x-axis from the center for a horizontal major axis. Therefore, the vertices are located at \((-4-4, 2) = (-8, 2)\) and \((-4+4, 2) = (0, 2)\).

Step 4: Calculate and Locate the Foci

The foci are located along the major axis. Calculate \(c\) using \(c^2 = a^2 - b^2\). Hence, \(c^2 = 16 - 9 = 7\) so \(c = \sqrt{7}\). The foci, therefore, are \((-4 - \sqrt{7}, 2)\) and \((-4 + \sqrt{7}, 2)\).

Step 5: Draw the Ellipse

With the center at \((-4, 2)\), vertices at \((-8, 2)\) and \((0, 2)\), and foci at \((-4 - \sqrt{7}, 2)\) and \((-4 + \sqrt{7}, 2)\), sketch an ellipse that is wider along the x-axis than it is tall along the y-axis. This ellipse should touch the vertices and be symmetric about the center.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. These geometric shapes include circles, ellipses, parabolas, and hyperbolas. Each type of conic section has unique properties and equations.
Ellipses, in particular, are often found in planetary orbits and optical systems. Their shape results from the intersection of a plane with a cone at an angle less steep than the cone's side but not parallel to its base.

• A circle is a special type of ellipse where both axes are of equal length.
• Ellipses have two axes, the major and minor, with different lengths.
• The general equation for a conic section is a quadratic one.

Understanding conic sections is crucial in mathematics and physics because they model various real-world phenomena.

Graphing Ellipses

Graphing an ellipse starts with understanding its standard form equation: \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]Here, \((h, k)\) is the center, and \(a\) and \(b\) are the semi-major and semi-minor axes respectively. Knowing these values lets you plot the ellipse on a coordinate plane.
When graphing:

• Identify the center of the ellipse from \((h, k)\).
• Determine the lengths of the semi-major (\(a\)) and semi-minor (\(b\)) axes.
• Calculate the full lengths of the major and minor axes by doubling \(a\) and \(b\), respectively.
• Mark the vertices and endpoints of the minor axis from the center.

In our example, the equation \[ \frac{(x+4)^{2}}{16} + \frac{(y-2)^{2}}{9} = 1 \]indicates that the ellipse is horizontally oriented with a center at \((-4, 2)\), and has vertices that define its shape and placement on the graph.

Foci and Vertices

Ellipses have special points called foci (singular: focus) and vertices that help define their shape. The vertices are points where the ellipse intersects its axes, while the foci are used to determine the curve's exact shape.
The calculation involves:

• Vertices: Positioned along the major axis at a distance of \(a\) from the center.
• Foci: Located along the major axis, at a distance of \(c\) from the center, calculated using \(c^2 = a^2 - b^2\).

In our example, the ellipse's vertices are at \((-8, 2)\) and \((0, 2)\). The foci, calculated using \(c = \sqrt{7}\), are at \((-4-\sqrt{7}, 2)\) and \((-4+\sqrt{7}, 2)\). These points provide insight into the ellipse's symmetrical nature and its orientation on a graph.