Problem 8 - Calculus Early Transcendentals

Question

What is the equation of the standard hyperbola with vertices at \((0, \pm a)\) and foci at \((0, \pm c) ?\)

Step-by-Step Solution

Answer

Answer: The equation of the standard hyperbola with vertices at (0, ±a) and foci at (0, ±c) is \[\frac{y^2}{a^2} - \frac{x^2}{(c^2 - a^2)} = 1\] with the hyperbola opening vertically.

Step 1: Recall the equations of a standard hyperbola

A standard hyperbola has two forms: \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\] if the hyperbola opens horizontally, and \[\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\] if the hyperbola opens vertically. Since the vertices and foci are given in the form \((0, \pm a)\) and \((0, \pm c)\) respectively, the hyperbola is opening vertically.

Step 2: Determine the values of 'a' and 'c'

The vertices are given as \((0, \pm a)\), which tells us that the distance from the center of the hyperbola to each vertex is equal to 'a'. Similarly, the foci are given as \((0, \pm c)\), which means the distance from the center of the hyperbola to each focus is equal to 'c'.

Step 3: Relate eccentricity, vertices, and foci

For a hyperbola, the relationship between the eccentricity (e), the distance from the center to the vertex (a), and the distance from the center to the focus (c) is given by: \[c = ae\] Since we know the values of 'a' and 'c', we can find 'e' using this relationship.

Step 4: Find the relationship between 'a' and 'b'

Using the relationship between the eccentricity (e), the distance from the center to the vertex (a), and the distance from the center to the focus (c), we can find the relationship between 'a' and 'b'. Since \[c = ae\], we know that \[e = \frac{c}{a}\]. Using the Pythagorean relation for hyperbola, \[a^2 + b^2 = c^2\], we can substitute the value of 'e' in the equation: \[a^2 + b^2 = (\frac{c}{a} * a)^2\]

Step 5: Write the equation for the hyperbola

With the given values for 'a' and 'c', substitute into the equation for the standard hyperbola opening vertically: \[\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\] Substituting the relationship between 'a' and 'b' found in the previous step, we get: \[\frac{y^2}{a^2} - \frac{x^2}{(c^2 - a^2)} = 1\] This is the equation of the standard hyperbola with vertices at \((0, \pm a)\) and foci at \((0, \pm c)\).

Key Concepts

Vertices of Hyperbola

The vertices of a hyperbola play a crucial role in defining its shape and orientation. For the type of hyperbola in the exercise, the vertices are located at the points \((0, \pm a)\), which tells us the hyperbola opens vertically. This implies that the major axis of the hyperbola is along the vertical direction.

The distance from the center of the hyperbola (which is generally at the origin \((0,0)\) in standard equations) to each vertex is 'a'.
This value 'a' helps determine how "spread out" or "narrow" the hyperbola appears.

Always remember:

• The vertices are always located "a" units away from the center along the axis of symmetry.
• If the vertices were at \((\pm a, 0)\), it would indicate a horizontal opening.

Understanding the placement of vertices is essential for crafting the correct equation and visually graphing the hyperbola.

Foci of Hyperbola

The foci are another essential component in understanding hyperbolas. In hyperbolas, the foci are two fixed points located along the major axis of the hyperbola.
For our vertical hyperbola, given the points of the foci are \((0, \pm c)\), they determine the positions of the foci along the vertical axis.

The value 'c' is the distance from the center to each focus.
This value is always greater than 'a', meaning that the foci are always outside the vertices for hyperbolas.

Important points to remember:

• The foci help determine the shape and curvature of the hyperbola.
• They are used in calculating the eccentricity of the hyperbola, which we'll discuss next.

Placement of the foci essentially helps in defining the behavior of the hyperbola's arms as they spread outward.

Eccentricity

Eccentricity is a fundamental concept that describes how "stretched" a conic section is compared to a circle.
For hyperbolas, the eccentricity 'e' is a measure of how much the conic section deviates from being circular.
It is given by the formula:
\[ e = \frac{c}{a} \]
Where:

• 'c' is the distance from the center to a focus.
• 'a' is the distance from the center to a vertex.

A higher eccentricity value indicates the hyperbola is more elongated, while a lower value means it is less pronounced.

Key characteristics:

• For hyperbolas, the eccentricity value is always greater than 1 (\(e > 1\)).
• The distance between foci plays a vital role in defining the eccentricity.

Eccentricity aids in understanding the geometry of the hyperbola and how it might differ from other conic sections such as ellipses and circles.