Problem 2 - Precalculus: Mathematics for Calculus

Question

The graph of the equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) with \(a>b>0\) is an ellipse with vertices (________ , ________ )and (_______ ,________ ) and foci \((\pm c, 0),\) where \(c=\) ________ . So the graph of \(\frac{x^{2}}{5^{2}}+\frac{y^{2}}{4^{2}}=1\) is an ellipse with vertices (_______ , _______ ) and (_______,_______) and foci (_______ , _______ ) and (_______,_______).

Step-by-Step Solution

Answer

Vertices: (5,0), (-5,0); Foci: (3,0), (-3,0).

Step 1: Identify Values of 'a' and 'b'

For the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the ellipse has semi-major axis \(a\) and semi-minor axis \(b\). From \(\frac{x^2}{5^2} + \frac{y^2}{4^2} = 1\), we identify \(a = 5\) and \(b = 4\).

Step 2: Find the Vertices

For an ellipse, the vertices along the major axis (horizontal in this case) are \((\pm a, 0)\). Thus, the vertices for our ellipse are \((\pm 5, 0)\), which translates to \((5,0)\) and \((-5,0)\).

Step 3: Calculate the Foci

The foci of an ellipse can be calculated using the formula \(c = \sqrt{a^2 - b^2}\). Here, \(c = \sqrt{5^2 - 4^2} = \sqrt{25 - 16} = \sqrt{9} = 3\).

Step 4: Determine the Foci Coordinates

The coordinates of the foci, given the ellipse is horizontal, are \((\pm c, 0)\). Therefore, the foci are \((3, 0)\) and \((-3, 0)\).

Key Concepts

Vertices of an Ellipse

Vertices are crucial points on an ellipse, as they define the shape and orientation of the ellipse. In the standard equation of an ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]where \(a > b > 0\), the vertices are located at a distance 'a' from the center along the major axis. When the major axis is horizontal, the vertices will be at

• \((a, 0)\)
• \((-a, 0)\)

This results in

• \((5, 0)\)
• \((-5, 0)\)

for an ellipse defined by the equation \(\frac{x^2}{5^2} + \frac{y^2}{4^2} = 1\).
These points are referred to as the endpoints of the major axis and represent the widest span of the ellipse.
Understanding the location of the vertices helps in visualizing and sketching the ellipse.

Foci of an Ellipse

The foci (or focuses) of an ellipse are the points fixed along the major axis from which the sum of the distances to any point on the ellipse is constant. In mathematical terms, for the ellipse given by the equation: \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]the foci can be found using the formula: \[c = \sqrt{a^2 - b^2}\]Plugging in the values from our specific ellipse equation \(\frac{x^2}{5^2} + \frac{y^2}{4^2} = 1\), we have:

• \(c = \sqrt{5^2 - 4^2} = \sqrt{25 - 16} = \sqrt{9} = 3\)

This tells us that the foci for our ellipse are located at

• \((3, 0)\)
• \((-3, 0)\)

These foci are important in understanding how the ellipse's shape is related to its definition, as every point on the ellipse will have an equal sum of distances to these two points.

Semi-Major Axis

The semi-major axis is one-half of the longest diameter of an ellipse. In simpler terms, it is the distance from the center of the ellipse to one of its vertices along the major axis.
In the context of the ellipse defined by \[\frac{x^2}{5^2} + \frac{y^2}{4^2} = 1\]the semi-major axis is determined by the larger denominator in the equation, which is \(5\) in this case.
Therefore, the semi-major axis

• Runs horizontally from \(-5\) to \(5\)
• Provides crucial information about the length and orientation of the ellipse

By knowing the length of the semi-major axis, you can easily visualize the width of the ellipse across its longest stretch, and with it being aligned along the x-axis here, it indicates the ellipse is wider horizontally than vertically. This width affects everything from plotting the ellipse to understanding the distribution of points, making the semi-major axis a key component in the ellipse's geometry.