Problem 26 - Algebra and Trigonometry with Analytic Geometry

Question

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Vertices \(V(\pm 4,0)\), passing through \((8,2)\)

Step-by-Step Solution

Answer

Equation: \(\frac{x^2}{16} - \frac{y^2}{2} = 1\).

Step 1: Identify the Hyperbola's Orientation

Since the vertices are at \((\pm 4,0)\), the hyperbola is horizontally oriented. The equation takes the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

Step 2: Determine \(a\)

The vertices are \((\pm 4,0)\), so \(a = 4\). Hence, \(a^2 = 16\). The equation becomes \(\frac{x^2}{16} - \frac{y^2}{b^2} = 1\).

Step 3: Use the Given Point to Find \(b^2\)

Substitute point \((8, 2)\) into the equation: \(\frac{8^2}{16} - \frac{2^2}{b^2} = 1\). Simplify the terms: \(\frac{64}{16} - \frac{4}{b^2} = 1\).

Step 4: Solve for \(b^2\)

\(\frac{64}{16} = 4\). So the equation becomes \(4 - \frac{4}{b^2} = 1\). Solve it: \(4 - 1 = \frac{4}{b^2}\), hence \(b^2 = 2\).

Step 5: Write the Hyperbola's Equation

Using \(a^2 = 16\) and \(b^2 = 2\), the equation of the hyperbola is \(\frac{x^2}{16} - \frac{y^2}{2} = 1\).

Key Concepts

Equation of a Hyperbola

The equation of a hyperbola provides the foundational structure that defines its shape and orientation. A hyperbola typically takes one of two forms, depending on its orientation:

• Horizontally oriented hyperbola: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
• Vertically oriented hyperbola: \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\)

For our exercise, since the vertices are at \( (\pm 4, 0) \), we identify it as a horizontally oriented hyperbola. This is why we use the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \).
By substituting the necessary values, such as those of \(a^2\) and \(b^2\), we can define the exact equation of the hyperbola, which in this case becomes \(\frac{x^2}{16} - \frac{y^2}{2} = 1\). This equation represents the mathematical relationship between \(x\) and \(y\) that all points on the hyperbola satisfy.

Vertices of a Hyperbola

The vertices of a hyperbola are crucial points that lie on the hyperbola's transverse axis. They define the distance from the center to the tips of the hyperbola. In the horizontal orientation, these points are given by \((\pm a, 0)\). For our specific case:

• The vertices are at \((\pm 4, 0)\), which implies that \(a = 4\).
• Since \(a\) is the distance from the center to each vertex, we calculate \(a^2 = 16\).

These vertices not only define the extent of the hyperbola along the x-axis but also help in determining the basic form of the hyperbola's equation.
The position of the vertices is directly influential in plotting the hyperbola on a Cartesian coordinate system.

Center of a Hyperbola

The center of a hyperbola is the midpoint from which the transverse and conjugate axes extend. In mathematical terms, the center of the hyperbola looks like where both axes cross, and it is symmetric to both parts of the hyperbola.
In this exercise, the center is at the origin, represented by the point \((0, 0)\).

• This point is crucial because all calculations involving the hyperbola rely on it as a reference mark.
• The center does not shift in this exercise, so the transverse axis along the horizontal line connects the vertices symmetrically around this point.

The position of the center helps to determine how and where the hyperbola is oriented, providing a base for establishing symmetry.

Orientation of a Hyperbola

The orientation of a hyperbola refers to the direction in which it opens. A hyperbola can be either horizontally or vertically oriented, which is become evident from where its vertices are placed.

• If the vertices are on the x-axis, the hyperbola is horizontally oriented.
• If the vertices are on the y-axis, the hyperbola is vertically oriented.

For our example, since the vertices are \((\pm 4, 0)\), the hyperbola's orientation is horizontal. This means that the hyperbola opens left and right, symmetrically about the horizontal axis.
Understanding the orientation is key to solving problems, as it dictates the structure of the hyperbola's equation. This orientation also affects how we solve for unknowns, like \(b^2\), by using known points on the hyperbola.