Problem 31 - Algebra and Trigonometry

Question

Find an equation for the hyperbola that satisfies the given conditions. Foci: \(( \pm 5,0),\) vertices: \(( \pm 3,0)\)

Step-by-Step Solution

Answer

The equation of the hyperbola is \(\frac{x^2}{9} - \frac{y^2}{16} = 1\).

Step 1: Identify the Center and Orientation

Since the foci are \(( \pm 5,0)\) and the vertices are \(( \pm 3,0)\), the hyperbola is centered at the origin and is oriented horizontally because the x-coordinates of the vertices and foci are non-zero while the y-coordinates are zero.

Step 2: Determine the Values of a and c

The distance from the center to each vertex is \(|\pm 3 - 0| = 3\). Thus, \(a = 3\). The distance from the center to each focus is \(|\pm 5 - 0| = 5\). Thus, \(c = 5\).

Step 3: Calculate the Value of b

For a hyperbola, the relationship between \(a\), \(b\), and \(c\) is given by the equation \(c^2 = a^2 + b^2\). Substitute the known values: \(5^2 = 3^2 + b^2\). This simplifies to \(25 = 9 + b^2\). Therefore, \(b^2 = 16\) and \(b = 4\).

Step 4: Write the Equation of the Hyperbola

The standard form of the equation for a hyperbola with a horizontal transverse axis is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). Substitute \(a = 3\) and \(b = 4\) into the equation: \(\frac{x^2}{9} - \frac{y^2}{16} = 1\).

Key Concepts

Conic Sections

Conic sections are shapes created by the intersection of a plane with a cone. Depending on the angle and position of the intersection, we can derive different forms:

• Circle - When the intersection is perpendicular to the cone's axis.
• Ellipse - When the intersection is at an angle, but does not pass through the base.
• Parabola - When the plane is parallel to a generating line of the cone.
• Hyperbola - When the plane intersects both halves of the cone.

A hyperbola consists of two separate curves, known as branches. Each branch is a mirror image of the other. Hyperbolas are unique due to the open nature of these curves, which extend indefinitely and never close.

Foci and Vertices

The foci and vertices are key points in a hyperbola that help us determine its shape and position.

• The foci are two fixed points located such that the difference of the distances from any point on the hyperbola to these foci is a constant. In this case, the foci are \(( \pm 5,0)\), meaning they are located at 5 units on either side of the origin along the x-axis.
• The vertices are points where each branch of the hyperbola is closest to each other. The given vertices are \((\pm 3,0)\), indicating that they are positioned at 3 units on either side of the origin.

These points are crucial for determining the orientation and the standard form of a hyperbola's equation.

Standard Form of a Hyperbola

The standard form of a hyperbola's equation depends on its orientation:

• Horizontal orientation has the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). For our given problem, the x-coordinates of the foci and vertices confirm that the hyperbola is horizontal. This form is used when the transverse axis is along the x-axis.
• Vertical orientation is expressed as \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \). This form is used when the transverse axis is along the y-axis.

The equation represents the relationship between the coordinates of any point on the hyperbola and the hyperbola's dimensions. Substituting the values of \(a\) and \(b\), gives us the specific equation of the hyperbola under consideration.

Values of a, b, and c

The values of \(a\), \(b\), and \(c\) are integral to defining a hyperbola:

• \(a\) - Represents the distance from the center to each vertex along the transverse axis. For our hyperbola, \(a = 3\), as derived from the distance from the center (0,0) to the vertices (\(\pm 3,0\)).
• \(c\) - Indicates the distance from the center to each focus. Here, \(c = 5\), matching the distance from the center to the foci (\(\pm 5,0\)).
• \(b\) - Calculated using the relationship \(c^2 = a^2 + b^2\). For this hyperbola, solving \(25 = 9 + b^2\) tells us that \(b = 4\).

These values not only help in crafting the equation but also provide insight into the geometry and dimensions of the hyperbola.