Problem 12 - Precalculus Mathematics for Calculus

Question

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$4 x^{2}+25 y^{2}=100$$

Step-by-Step Solution

Answer

Vertices: \((5,0), (-5,0)\); Foci: \((\sqrt{21},0), (-\sqrt{21},0)\); Eccentricity: \(\frac{\sqrt{21}}{5}\); Major axis: 10, Minor axis: 4.

Step 1: Rewrite the Ellipse Equation in Standard Form

Given the equation \(4x^2 + 25y^2 = 100\). First, divide every term by 100 to get \(\frac{x^2}{25} + \frac{y^2}{4} = 1\). This equation is now in the standard form of an ellipse centered at the origin.

Step 2: Identify the Orientation and Parameters

From \(\frac{x^2}{25} + \frac{y^2}{4} = 1\), we identify that \(a^2 = 25\) and \(b^2 = 4\). Since \(a^2 > b^2\), the major axis is along the x-axis. Thus, \(a = 5\) and \(b = 2\).

Step 3: Find the Vertices

The vertices are along the x-axis, given by \((\pm a, 0)\). Substitute \(a = 5\), so the vertices are \((5, 0)\) and \((-5, 0)\).

Step 4: Find the Foci

The distance of the foci from the center is \(c\) where \(c^2 = a^2 - b^2\). Calculate \(c^2 = 25 - 4 = 21\), so \(c = \sqrt{21}\). The foci are \((\pm \sqrt{21}, 0)\).

Step 5: Determine the Eccentricity

The eccentricity \(e\) of the ellipse is given by \(e = \frac{c}{a}\). Substituting the values, \(e = \frac{\sqrt{21}}{5}\).

Step 6: Calculate the Lengths of the Axes

The major axis length is \(2a = 10\), and the minor axis length is \(2b = 4\).

Step 7: Sketch the Graph

Draw an ellipse centered at the origin with vertices at \((5, 0)\) and \((-5, 0)\) on the x-axis, and co-vertices at \((0, 2)\) and \((0, -2)\) on the y-axis. Include foci at \((\sqrt{21}, 0)\) and \((-\sqrt{21}, 0)\).

Key Concepts

Vertices of an Ellipse

In any ellipse, the vertices are the furthest points on either side along the major axis. They essentially define the span of the ellipse in its longest direction. For a given equation of an ellipse, the vertices can be found by identifying the value of \(a\), the distance from the center to each vertex along the major axis.

In our example, the ellipse's equation after being rewritten in standard form is \(\frac{x^2}{25} + \frac{y^2}{4} = 1\). Here, \(a^2 = 25\), meaning \(a = 5\). This tells us that our ellipse's vertices will be situated at \((\pm 5, 0)\) since the major axis is along the x-axis. These points denote the farthest reach of the ellipse horizontally.

Foci of an Ellipse

The foci of an ellipse are two fixed points situated inside the ellipse along the major axis. They play a crucial role as the sum of the distances from any point on the ellipse to the foci is constant. This peculiar property helps in defining the shape of an ellipse.

To locate the foci, we use the equation \(c^2 = a^2 - b^2\), where \(c\) is the distance from the center to each focus. In this ellipse, substituting the given values, \(c^2 = 25 - 4 = 21\), and so \(c = \sqrt{21}\). Thus, the foci are located at \((\pm \sqrt{21}, 0)\) since the major axis is on the x-axis. The closer the foci are to the center, the more circular the ellipse appears.

Eccentricity of an Ellipse

Eccentricity is a key feature of an ellipse that defines its roundness or flatness. It is denoted by \(e\) and can be calculated using the formula \(e = \frac{c}{a}\), where \(c\) is the distance from the center to a focus, and \(a\) is the distance from the center to a vertex.

In our example, with \(c = \sqrt{21}\) and \(a = 5\), we find \(e = \frac{\sqrt{21}}{5}\). The value of eccentricity for an ellipse always lies between 0 and 1. A value closer to 0 means the ellipse is more circular, while a value closer to 1 indicates a more elongated shape.

Major Axis

The major axis of an ellipse is the longest diameter that passes through the center and both foci of the ellipse. It also includes the vertices and is aligned along the direction where the ellipse stretches the furthest.

For the equation \(\frac{x^2}{25} + \frac{y^2}{4} = 1\), the major axis is aligned along the x-axis as \(a^2 > b^2\). Since \(a = 5\), the total length of the major axis is \(2a = 10\). This measurement spans from vertex to vertex, encompassing the entire length of the ellipse.

Minor Axis

The minor axis is the shortest diameter of the ellipse and runs perpendicular to the major axis through the center. It represents the ellipse's narrowest dimension.

In our example, from the equation \(\frac{x^2}{25} + \frac{y^2}{4} = 1\), we see that \(b^2 = 4\) hence \(b = 2\). This means the minor axis, running along the y-axis, has a total length of \(2b = 4\). This axis forms the shorter span across the ellipse as it bisects it from co-vertex to co-vertex.