Problem 39 - Precalculus: Mathematics for Calculus

Question

Find an equation for the hyperbola that satisfies the given conditions. Foci: \((0, \pm 2),\) vertices: \((0, \pm 1)\)

Step-by-Step Solution

Answer

The equation is \(y^2 - \frac{x^2}{3} = 1\).

Step 1: Understanding the Hyperbola

The given hyperbola has its foci on the y-axis at points \((0, \, \pm 2)\) and vertices at points \((0, \, \pm 1)\). This suggests that it is a vertical hyperbola centered at the origin \((0, 0)\).

Step 2: Identify Parameters 'a' and 'c'

For a hyperbola, the coordinates of the vertices \((0, \, \pm a)\) provide 'a'. Here, \(a = 1\). The foci have coordinates \((0, \, \pm c)\), giving us \(c = 2\).

Step 3: Calculate Parameter 'b'

Use the relationship for hyperbolas: \(c^2 = a^2 + b^2\). Substituting the known values, \(4 = 1 + b^2\). Thus, \(b^2 = 3\).

Step 4: Write the Hyperbola Equation

For a vertical hyperbola centered at the origin, the equation is \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\). Using the identified values \(a^2 = 1\) and \(b^2 = 3\), the equation becomes \(\frac{y^2}{1} - \frac{x^2}{3} = 1\) or simplified to \(y^2 - \frac{x^2}{3} = 1\).

Key Concepts

Vertical Hyperbola

When dealing with hyperbolas, it's essential to determine whether the hyperbola is vertical or horizontal. A vertical hyperbola opens up and down along the y-axis, meaning its primary features, like foci and vertices, are aligned vertically. In contrast, a horizontal hyperbola opens left and right along the x-axis. The sign difference between these two is crucial, as it dictates the alignment and orientation of the hyperbola.
In our problem, the foci are given at \((0, \, \pm 2)\) and vertices at \((0, \, \pm 1)\). This tells us the hyperbola is vertical, centered at the origin. Visualizing this, imagine two arms extending from the top and bottom of the center, accentuating that this is unlike its horizontal sibling, which has a broader left-right span.

• The confirmation of a vertical hyperbola is due to the y-coordinates having the \(\pm\) values.
• Its representation is essential as it directs us to the correct form of the hyperbola equation.

Foci and Vertices

The foci and vertices of a hyperbola are key features that define its shape and orientation. The vertices \((0, \, \pm 1)\) are the points where the hyperbola crosses the axis of symmetry, located on the y-axis due to its vertical nature. These are significant as they provide direct insight into the dimension 'a', the distance from the center to either vertex, which here is 1.

The foci, \((0, \, \pm 2)\), though not actually on the hyperbola, are integral to its definition. They reside inside the arms of the hyperbola and describe how "spread out" it is. In our case, 'c', the distance from the center to a focus, is 2.

• Vertices mark the closest approach and farthest distance along the main axis.
• The location of foci relates to the spread and openness of the hyperbola branches.

Hyperbola Parameters

Hyperbola parameters such as `a`, `b`, and `c` are not just numbers; they are geometrically and algebraically fundamental to describing the hyperbola. The parameter 'a' represents the distance from the center to each vertex. Here, \(a = 1\).

The parameter 'c' is the distance from the center to each focus, defined as \(c = 2\) here. The relationship between these is captured in the equation \(c^2 = a^2 + b^2\). Substituting the known values yields:

• \(4 = 1 + b^2\)

This equation allows us to solve for the third parameter 'b', representing the distance associated with the conjugate axis of the hyperbola. Solving gives \(b^2 = 3\).

Hyperbola Formula

Once all parameters are understood, we can construct the hyperbola's equation, which gives a complete idea of its geometric structure. For vertical hyperbolas, the standard formula is:

• \[\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\]

Given our identified parameters, where \(a^2 = 1\) and \(b^2 = 3\), the equation simplifies to:\[y^2 - \frac{x^2}{3} = 1\]This formula precisely outlines the shape and size of the hyperbola centered at the origin. It also guides us in graphing the hyperbola and understanding how its arms open and stretch with respect to its center. The simplicity of this equation belies the complex relationships between its geometric features.