Problem 3 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the parameters

The equation of the ellipse is \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \) with \( a > b > 0 \). Given \( \frac{x^2}{4^2} + \frac{y^2}{5^2} = 1 \), we identify \( a = 5 \) and \( b = 4 \).

Step 2: Determine the vertices

For an ellipse with a vertical major axis, the vertices are located at \((0, a)\) and \((0, -a)\). Since \( a = 5 \), the vertices are at \((0, 5)\) and \((0, -5)\).

Step 3: Calculate c

The formula to find \( c \) for an ellipse is \( c = \sqrt{a^2 - b^2} \). Substituting \( a = 5 \) and \( b = 4 \), we find \( c = \sqrt{5^2 - 4^2} = \sqrt{25 - 16} = \sqrt{9} = 3 \).

Step 4: Determine the foci

For an ellipse with a vertical major axis, the foci are located at \((0, c)\) and \((0, -c)\). With \( c = 3 \), the foci are at \((0, 3)\) and \((0, -3)\).

Key Concepts

ellipse vertices

When studying ellipses, a key feature is their vertices, which are the endpoints of the major axis. For the ellipse equation given in the problem, we have:

• The form of the equation suggests it is vertical since the larger denominator corresponds to the \(y\) variable.
• For ellipses with a vertical major axis, the vertices are positioned at \( (0, a) \) and \( (0, -a) \).

Since \(a = 5\), the vertices of this particular ellipse are at \( (0, 5) \) and \( (0, -5) \). These points signify the maximum extent of the ellipse along the vertical direction. Therefore, identifying the vertices plays a critical role in forming a visual understanding of the ellipse.

ellipse foci

In an ellipse, the foci are two special points that lie along the major axis. Unlike circles, where the center is the only focal point, ellipses have this pair which influences their shape:

• For a vertical ellipse, the foci are vertically aligned at \( (0, c) \) and \( (0, -c) \).
• The value \( c \) is a crucial parameter that needs to be calculated using the relationship \( c = \sqrt{a^2 - b^2} \).

With \(a = 5\) and \(b = 4\), calculating gives \( c = \sqrt{25 - 16} = \sqrt{9} = 3 \). Thus, the foci are located at \( (0, 3) \) and \( (0, -3) \). The presence of the foci ensures that the sum of distances from any point on the ellipse to the foci is constant, which is the essential property of ellipses.

ellipse parameters

To fully describe an ellipse, specific parameters must be understood. These elements differentiate one ellipse from another and give its unique shape and size:

• \(a\): the semi-major axis, representing half of the longest diameter of the ellipse.
• \(b\): the semi-minor axis is half of the shortest diameter, affecting the ellipse's width.

In our example, this translates to \(a = 5\) and \(b = 4\). These parameters indicate that the ellipse stretches more in the vertical direction. Understanding these parameters helps you sketch the ellipse and determine its properties accurately, such as orientation and eccentricity. Thus, even if the equation format changes with different coefficients, you can always locate the vertices and foci by simply recalculating these vital parameters.