Problem 24 - Algebra 2

Question

Write the equation of a hyperbola with the given foci and vertices. foci \((0, \pm 13),\) vertices \((0, \pm 5)\)

Step-by-Step Solution

Answer

The equation of the hyperbola with the given foci \((0, \pm 13)\), and vertices \((0, \pm 5)\) is \(\frac{y^2}{5^2} - \frac{x^2}{12^2} = 1\).

Step 1: Determine the lengths of semi-major and semi-minor axes

The vertices of the hyperbola are given as \((0, \pm 5)\), so we determine the length of the semi-major axis 'a' to be 5. This is obtained by subtracting the y-components of the vertices. Similarly, the foci are given by \((0, \pm 13)\), the focal distance 'c' is 13.

Step 2: Determine the value of 'b' using the hyperbola equation

In a hyperbola, the relationship between 'a', 'b' and 'c' is given by \(c = \sqrt{a^2 + b^2}\). Therefore, \(b = \sqrt{c^2 - a^2}\) which gives \(b = \sqrt{13^2 - 5^2} = 12\).

Step 3: Write down the equation

We already identified the hyperbola is centered at the origin and its transverse axis is the y-axis since the vertices and foci were given in terms of y. Therefore the equation can be written in the standard form \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\). Substituting for the value of 'a' and 'b', gives us the equation \(\frac{y^2}{5^2} - \frac{x^2}{12^2} = 1\).

Key Concepts

Semi-major axis

The semi-major axis of a hyperbola is the distance from the center of the hyperbola to a vertex along the line that splits the hyperbola symmetrically. In the given exercise, the vertices are located at \(0, \pm 5\), which means that the center of the hyperbola is at the origin \(0, 0\). Since the vertices are along the y-axis, the semi-major axis is vertical.

In this example, the length of the semi-major axis is 5. This is because the distance from the center (0,0) to either vertex is 5 units. The notation used to represent this length is usually 'a,' such that \(a=5\).

• The semi-major axis determines the opening of the hyperbola.
• It is an important parameter for writing the equation of a hyperbola.

Understanding the semi-major axis helps in graphing and defining the hyperbola, as it gives clarity on how wide each branch of the hyperbola extends along its transverse axis.

Semi-minor axis

The semi-minor axis of a hyperbola is the distance from the center of the hyperbola to the edge along the conjugate axis. Unlike the semi-major axis, it is not the longest radius but plays a crucial role in constructing the hyperbola's equation. In our exercise, we calculated 'b' using the relationship between the axes and the foci.

The relationship between the semi-major axis \(a\), the semi-minor axis \(b\), and the focal distance \(c\) is given by the formula \(c = \sqrt{a^2 + b^2}\).

For this hyperbola, we were given \(c=13\) and \(a=5\), allowing us to solve for \(b\) using \(b=\sqrt{c^2-a^2}\). This calculation shows that \(b=12\).

• The semi-minor axis does not affect the direction of the opening, but changes the hyperbola's shape.
• Its length contributes to the correct proportions when plotting the hyperbola on the Cartesian plane.

Understanding both axes is essential to accurately graph and comprehend the hyperbola's behavior.

Standard form of hyperbola

The standard form of a hyperbola is a specific mathematical representation that allows us to easily identify its characteristics by inspecting its equation. For hyperbolas centered at the origin, this form changes depending on the transverse axis. It helps us understand how the graph is oriented on the Cartesian coordinate plane.

For a vertical transverse axis, like in the given exercise, the standard form is \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\).

Here, the coefficients underneath \(y^2\) and \(x^2\) are important:

• 'a' is the semi-major axis - the bigger value under the component associated with the transverse axis direction.
• 'b' is the semi-minor axis - the component along the conjugate axis.

Utilizing this formula, we substitute \(a = 5\) and \(b = 12\) as identified in our solution. Therefore, the equation becomes \(\frac{y^2}{5^2} - \frac{x^2}{12^2} = 1\). Knowing this form allows us to write and manipulate hyperbola equations precisely and analytically.

Vertices

Vertices of a hyperbola are the points where each branch of the hyperbola is closest to its center. These positions are crucial to understanding the hyperbola's shape and position. In the exercise, the vertices are given as \(0, \pm 5\), indicating they lie on the y-axis at 5 units above and below the center, which is at the origin.

The vertices define the direction of the hyperbola's opening.

In the given hyperbola:

• Since vertices are aligned along the y-axis, it signifies a vertical transverse axis.
• This determines that the hyperbola opens upwards and downwards.

Knowing the vertices not only assists in plotting the graph but also plays a key role in forming the hyperbola's equation. They act as a direct visual and numerical guide to the hyperbola's largest spread.

Foci

The foci are the two points from which the sum of the distances to any point on the hyperbola is constant. For the given exercise, the foci are located at \(0, \pm 13\). These points lie on the y-axis, indicating a vertical orientation consistent with the vertices.

The distance from the center to a focus is denoted by 'c' and is a defining feature of a hyperbola:

• It is used in the relationship \(c = \sqrt{a^2+b^2}\).
• It affects the 'spread' of the hyperbola, showing how 'stretched' it becomes along its transverse axis.

Since \(c = 13\) in this case, the foci lie further from the center than the vertices, which is always true in hyperbolas. The position and calculation of the foci assist in forming the complete picture of the hyperbola's geometry.