Problem 278 - Calculus Volume 3

Question

Determine the equation of the hyperbola using the information given. (0, 10), (0, ?10) and eccentricity of 2.5

Step-by-Step Solution

Answer

The equation of the hyperbola is \(\frac{y^2}{100} - \frac{x^2}{525} = 1\).

Step 1: Identify the Vertices

We are provided with the points \((0, 10)\) and \((0, -10)\) which represent the vertices of the hyperbola. This means the hyperbola is vertical, and its center is at the origin \((0, 0)\). The distance between the center and each vertex is known as \(a\), hence \(a = 10\).

Step 2: Calculate the Distance from Center to Foci

The eccentricity \(e\) of a hyperbola is given by the formula \(e = \frac{c}{a}\), where \(c\) is the distance from the center to each focus. We have \(e = 2.5\) and \(a = 10\). Solving \(2.5 = \frac{c}{10}\), we find \(c = 2.5 \cdot 10 = 25\).

Step 3: Use the Relationship Between a, b, and c

For hyperbolas, we have the relationship \(c^2 = a^2 + b^2\). Plugging in our values, \(25^2 = 10^2 + b^2\), or \(625 = 100 + b^2\). Solving for \(b^2\), we get \(b^2 = 525\).

Step 4: Write the Equation of the Hyperbola

Since the hyperbola is vertical, its standard form is \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\). Substituting \(a^2 = 100\) and \(b^2 = 525\), the equation becomes \(\frac{y^2}{100} - \frac{x^2}{525} = 1\).

Key Concepts

Eccentricity

The eccentricity (\( e \)) of a hyperbola helps us understand its shape. It's a measure of how much the conic section deviates from being circular. For hyperbolas, \( e \) is always greater than 1.
It is calculated using the formula \( e = \frac{c}{a} \), where \( c \) is the distance from the center to each focus, and \( a \) is the distance from the center to each vertex.

• An eccentricity of 2.5 means the hyperbola is fairly elongated.
• If it were closer to 1, the hyperbola would appear less stretched and more rounded.

Learning how eccentricity changes the shape of hyperbolas helps in visualizing and understanding their graphs clearly.

Vertices of a Hyperbola

Vertices are significant points in a hyperbola, situated along the transverse axis, which is the line passing through these points. In this exercise, the vertices are at \((0, 10)\) and \((0, -10)\).
Since both vertices lie on the y-axis, this hyperbola is vertical.
Identifying the direction (vertical or horizontal) is crucial because it dictates the form of the hyperbola's equation.
The distance from the center to the vertex is denoted as \(a\). Here, \(a = 10\).

• The main transverse axis length is \(2a = 20\)
• Horizontal orientation would shift the points to the x-axis, but in this case, the y-values are what matter.

Understanding vertices helps form the initial basis for the hyperbola's equation.

Distance from Center to Foci

The foci of a hyperbola are key points lying outside the vertex on the hyperbola's main axis. They are essential for defining the curve.
Using the given eccentricity, \(e = 2.5\), and the formula \(e = \frac{c}{a}\), we calculated that \(c = 25\).
This tells us the foci are \(25\) units away from the center, which invites deeper exploration into the hyperbola's breadth.

• Foci are located at \((0, 25)\) and \((0, -25)\).
• This placement impacts the hyperbola's openness and stretch.

Recognizing where the foci are helps underline the definition of the hyperbola's spread apart structure.

Relationship Between a, b, and c in Hyperbolas

In hyperbolas, there is an intrinsic mathematical relationship between \(a\), \(b\), and \(c\), given by the equation \(c^2 = a^2 + b^2\).
Here, \(c = 25\), \(a = 10\), and we solve for \(b\). This relationship ensures that all components of the hyperbola are accurately represented.
Solving the equation \(625 = 100 + b^2\), we find that \(b^2 = 525\).

• The value of \(b\) holds implications for the hyperbola’s side-to-side stretch (conjugate axis).
• In this context, even though the hyperbola appears vertically oriented, \(b\) tells us about its outward stretch along the x-axis.

Balancing all three components, \(a\), \(b\), and \(c\), is crucial to define the complete equation of the hyperbola.