Problem 12 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (±3,0)\(;\) foci: (±6,0)

Step-by-Step Solution

Answer

The standard form of the equation of the hyperbola with the given vertices and foci is \(x^2/9 - y^2/27 = 1\).

Step 1: Identify the a value

In this problem, the vertices are at (±3,0), so a will be the absolute value of the x-coordinate of the vertex, which is 3.

Step 2: Identify the c value

Similarly, the foci are at (±6,0), so c will be the absolute value of the x-coordinate of the focus, which is 6.

Step 3: Find the b value

Using the relationship between a, b, and c for a hyperbola, which is \(c^2 = a^2 + b^2\), we can find the b value. We know the values of a as 3 and c as 6, substituting in the equation: \(6^2 = 3^2 + b^2 => b^2 = 36 - 9 => b^2 = 27\). Hence, b is \(sqrt{27}\), or 3\(sqrt{3}\).

Step 4: Write the standard form equation

Knowing that a = 3, b = 3\(sqrt{3}\), for a horizontal hyperbola the standard form of the equation will be \(x^2/3^2 - y^2/(3sqrt{3})^2 = 1\) or written more simply, \(x^2/9 - y^2/27 = 1\).

Key Concepts

Vertices

In the geometry of hyperbolas, vertices are important points that help define the shape of the curve. They represent where the hyperbola opens widest. For the given exercise, the vertices are positioned at (±3,0). This tells us that the hyperbola is centered at the origin and opens along the x-axis.

The distance from the center (0,0) to each vertex is called the value of 'a'. Here, this distance is 3 units, making the value of 'a' equal to 3. This also confirms that the transverse axis, which runs between the vertices, lies along the x-axis. Understanding the location and distance of the vertices is crucial for determining the other parameters of the hyperbola.

Foci

Hyperbolas have foci, which are special points located inside each arm of the hyperbola, along the same axis as the vertices. They help in defining the curve's shape and direction, and they have mathematical importance. In this case, the foci are located at (±6,0). This is further from the center than the vertices, indicating a specific property of hyperbolas: the foci are always positioned along the axis farther away from the center than the vertices.

The distance from the center to a focus is represented by the value 'c'. Here, 'c' equals 6. This value plays a vital role in determining the shape and form of the hyperbola, since together with 'a', it helps us find another important value, 'b', which further defines the curve's characteristics.

Standard Form

The standard form of a hyperbola’s equation is central to graphing and understanding its properties. There are two possible standard forms, depending on the hyperbola's orientation. For our horizontal hyperbola, the equation takes the form:

• \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)

Here, the important parameters are 'a' and 'b', which represent the distances mentioned above. In our solved exercise example, 'a' = 3, and through calculations using the relationship between 'a', 'b', and 'c' (i.e., \(c^2 = a^2 + b^2\)), we find that 'b' equals \(sqrt{27}\), or \(3\sqrt{3}\). These values plug directly into the equation, giving us the expression that details the hyperbola's geometric structure.

Equation of a Hyperbola

An equation of a hyperbola can be written once we identify the values of 'a' and 'b'. These equations differ slightly depending on the hyperbola's orientation (horizontal or vertical). Given that our specific problem deals with a horizontal hyperbola, the equation emerges clearly:

It is calculated as \(\frac{x^2}{9} - \frac{y^2}{27} = 1\), where '9' is \(3^2\) (because 'a' is 3), and '27' is \((3\sqrt{3})^2\). This formulation helps in graphically representing the hyperbola and understanding its documents based on its axis length and orientation.

Understanding the correct form of the hyperbola’s equation also helps when identifying key features such as asymptotes and interpreting real-world applications where hyperbolas play a role.