Problem 25 - Algebra 2

Question

Write the equation of a hyperbola with the given foci and vertices. foci \((0, \pm 2),\) vertices \((0, \pm 1)\)

Step-by-Step Solution

Answer

The equation of the hyperbola given the foci (0, ±2) and vertices (0, ±1) is \(y^2 - 3x^2 = 1\).

Step 1: Identify the values of 'a', 'b' and 'c'

In this problem, the foci are at points (0, ±2) and the vertices are at points (0, ±1). This tells us that the hyperbola is vertical because both the foci and vertices lie along the y-axis. The value of 'a' is the distance from the center to a vertex. Given the vertices are (0, ±1), 'a' equals 1. The value of 'c' is the distance from the center to a focus. Given the foci are (0, ±2), 'c' equals 2. We can find 'b' by using the equation \(c^2 = a^2 + b^2\) for a hyperbola. Plugging in the values we get \(b = \sqrt{c^2 - a^2} = \sqrt{2^2 - 1^2} = \sqrt{3}\)

Step 2: Write the equation of the hyperbola

Now we can write the equation of the hyperbola. The equation for a vertical hyperbola is \((\frac{y-h}{a})^2 - (\frac{x-k}{b})^2 = 1\). The center of our hyperbola is at the origin (0,0) so 'h' and 'k' are both 0. Plugging in the values we found for 'a' and 'b' we get \((\frac{y}{1})^2 - (\frac{x}{\sqrt{3}})^2 = 1\), which simplifies to \(y^2 - 3x^2 = 1\).

Key Concepts

Foci

The foci of a hyperbola are two specific points located along its axis of symmetry. For our specific hyperbola, these points have coordinates

• (0, 2)
• (0, -2)

The foci are critical because they help define the shape and orientation of the hyperbola. In general, the distance from any point on the hyperbola to each focus has the difference equal to a constant value. For our exercise, both foci are positioned along the y-axis, indicating it's a vertical hyperbola. This orientation plays a role in shaping the equation for the hyperbola, as it determines whether the 'y' or the 'x' component will lead the equation. The distance from the center to each focus is denoted by 'c', which in this case is 2.

Vertices

Vertices are two key points located on the hyperbola that represent the closest distance the curve reaches the center. In this specific problem, the vertices are given as

• (0, 1)
• (0, -1)

These vertices lie directly on the y-axis, signifying a vertical hyperbola. The vertices are instrumental in helping to define the size and position of the hyperbola on the graph. The distance from the center to either vertex is known as 'a'. For this exercise, 'a' is equal to 1, since the vertices are one unit away from the origin along the y-axis.

Equation of a Hyperbola

To write the equation of a hyperbola, especially when given its foci and vertices, it's essential to understand the standard form. For any hyperbola, the relationship between 'a', 'b', and 'c' is given by the equation \[c^2 = a^2 + b^2\]By solving for 'b' we found that \[b = \sqrt{3}\]For a vertical hyperbola, the standard form of the equation is \[\left( \frac{y-h}{a} \right)^2 - \left( \frac{x-k}{b} \right)^2 = 1\]where

• 'h' and 'k' are the coordinates of the center,
• 'a' is the distance from the center to a vertex,
• 'b' is derived from the relation with 'a' and 'c'.

In this exercise, 'h' and 'k' are both zero since the center is located at the origin (0,0). Using the values found for 'a=1' and 'b=\sqrt{3}', the equation of our hyperbola simplifies and becomes \[y^2 - 3x^2 = 1\]

Vertical Hyperbola

When we refer to a hyperbola as being 'vertical', it means its transverse axis (the line connecting the vertices) is aligned vertically, along the y-axis in this case. This particular orientation is the reason why the equation for our hyperbola is structured as \[\left( \frac{y}{a} \right)^2 - \left( \frac{x}{b} \right)^2 = 1\]with the 'y' term leading. The main characteristics of a vertical hyperbola include:

• Foci placed on the y-axis.
• Vertices lying on the y-axis.
• The equation having a leading 'y' component.

This differs from a horizontal hyperbola, in which the transverse axis runs along the x-axis. Therefore, recognizing the orientation—vertical vs horizontal—is key to correctly constructing the equation and understanding its graph.