Problem 5 - Precalculus: Mathematical 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. $$\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$$

Step-by-Step Solution

Answer

Vertices: (5, 0) and (-5, 0). Foci: (4, 0) and (-4, 0). Eccentricity: 0.8. Major axis length: 10, Minor axis length: 6.

Step 1: Identify the Form of the Ellipse Equation

The given ellipse equation is \( \frac{x^{2}}{25} + \frac{y^{2}}{9} = 1 \). This is in the standard form of \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \), where \( a^{2} = 25 \) and \( b^{2} = 9 \). Since \( a^{2} > b^{2} \), this is a horizontally oriented ellipse.

Step 2: Calculate the Length of the Major Axis

In a horizontal ellipse, the major axis is along the x-axis. The length of the major axis is given by \( 2a \). Since \( a^{2} = 25 \), we have \( a = \sqrt{25} = 5 \). Therefore, the major axis length is \( 2 \times 5 = 10 \).

Step 3: Calculate the Length of the Minor Axis

For a horizontal ellipse, the minor axis is along the y-axis. The length of the minor axis is given by \( 2b \). Since \( b^{2} = 9 \), we have \( b = \sqrt{9} = 3 \). Therefore, the minor axis length is \( 2 \times 3 = 6 \).

Step 4: Find the Vertices of the Ellipse

The vertices of the ellipse are located on the major axis, at a distance \( a \) from the center. Since this is a horizontal ellipse, the vertices are at \( (\pm a, 0) \). Thus the vertices are \( (5, 0) \) and \( (-5, 0) \).

Step 5: Calculate the Foci of the Ellipse

The foci for a horizontal ellipse are located at \( (\pm c, 0) \), where \( c = \sqrt{a^{2} - b^{2}} \). Calculating \( c \), we find \( c = \sqrt{25 - 9} = \sqrt{16} = 4 \). Thus, the foci are at \( (4, 0) \) and \( (-4, 0) \).

Step 6: Determine the Eccentricity of the Ellipse

The eccentricity \( e \) of an ellipse is given by \( e = \frac{c}{a} \). Using the values for \( c \) and \( a \), we find \( e = \frac{4}{5} = 0.8 \).

Step 7: Sketching the Ellipse

To sketch the ellipse, plot the center at the origin \((0,0)\). Mark the vertices at \((5,0)\) and \((-5,0)\), and the co-vertices (along the minor axis) at \((0, 3)\) and \((0, -3)\). Draw a smooth oval shape connecting these points. The foci \((4,0)\) and \((-4,0)\) indicate the internal focal points.

Key Concepts

Vertices of Ellipse

When discussing the ellipse equation \[ \frac{x^{2}}{25} + \frac{y^{2}}{9} = 1 \], an important part of understanding its geometry is identifying its vertices. Recall that vertices are special points where the ellipse intersects the major axis. To find them, observe that the major axis aligns horizontally since \( a^2 > b^2 \).

• Major axis length: \( 2a = 10 \)
• The vertices are at \((\pm a, 0)\)

For this ellipse, with \( a = 5 \), the vertices are located at \( (5, 0) \) and \( (-5, 0) \). These points are equidistant from the center, stretching along the line defined by the major axis, which is horizontally oriented.

Foci of Ellipse

In an ellipse, apart from the vertices, the foci are critical in defining its shape. For our ellipse, these are situated along the major axis. The distance from the center to a focus (denoted as \( c \)) is calculated using the formula:

• \( c = \sqrt{a^{2} - b^{2}} \)
• For our ellipse: \( c = \sqrt{25 - 9} \)
• Thus, \( c = 4 \)

This means that the foci are positioned at \((4,0)\) and \((-4,0)\). These points are located inside the ellipse, equidistant from the center, highlighting the axial symmetry of ellipses.

Eccentricity of Ellipse

The eccentricity of an ellipse is a measure of its deviation from being a perfect circle. For our horizontal ellipse:

• Eccentricity \( e \) is given by: \( e = \frac{c}{a} \)
• Substituting values, \( e = \frac{4}{5} = 0.8 \)

An eccentricity of 0 would imply a circle, and values between 0 and 1 describe ellipses. Since \( e = 0.8 \), our ellipse is significantly elongated, but not too stretched out.

Major and Minor Axes of Ellipse

Understanding the axes of an ellipse is crucial because they determine its size and shape.

• The major axis is the longest diameter: it runs horizontally in this ellipse because \( a^2 > b^2 \).
• Major axis length: \( 2a = 10 \)
• The minor axis is the shorter diameter: running vertically.
• Minor axis length: \( 2b = 6 \)

Together, the axes define a rectangle in which the ellipse fits snugly. This geometric property is key to sketching or visualizing ellipses.

Graph Sketching of Ellipse

Sketching an ellipse from its equation involves understanding its center, axes, and special points. The given ellipse's center is at the origin \((0,0)\). Here’s how you can sketch it:

• Start by marking the vertices, \((5,0)\) and \((-5,0)\), along the x-axis.
• Next, mark the co-vertices, which are on the y-axis: \((0, 3)\) and \((0, -3)\).
• Draw the semi-major and semi-minor axes to guide your sketch.
• Finally, draw a smooth, symmetrical oval shape through these points.
• Remember, the foci, \((4,0)\) and \((-4,0)\), aid in constructing the perfect ellipse drawing.

Here's a trick: imagine tugging a string attached to the foci to define the boundary of the ellipse visually. This exercise makes sketching straightforward and ensures it reflects the correct proportions.