Problem 10 - Algebra 2

Question

Find the equation of each hyperbola described. All points on the hyperbola are 72 units closer to one focus than the other. The foci are located at \((0,0)\) and \((300,0) .\)

Step-by-Step Solution

Answer

The equation of the given hyperbola is \( \frac{x^{2}}{1296} - \frac{y^{2}}{20736} = 1 \).

Step 1: Identifying the Foci and Asymptotes

The given coordinates indicate that the foci of the hyperbola are at the points (0,0) and (300,0). Therefore, it is an horizontal hyperbola since the foci are arranged horizontally. The distance between the foci is 300 units. So, half of that, or the distance from the center to each focus, is 150 units. The difference in the distances from any point on the hyperbola to the foci is 72 units. This is actually the value of 2a, where \(a\) is the distance from the center to a vertex. Hence, \(a=36\).

Step 2: Calculating the Value of b

The values of \(a\) and \(c\) (the distance from the center to the focus) are used to find the value of \(b\), the distance from the center to a unit point on the hyperbola. This is dictated by the relationship \(c^{2} = a^{2} + b^{2}\), where \(c=150\) and \(a=36\). Re-arranging the equation to solve for \(b\) we get \(b = \sqrt{c^{2} - a^{2}} = \sqrt{150^{2} - 36^{2}} = 144\).

Step 3: Writing Down the Equation of the Hyperbola

For a horizontal hyperbola centered at the origin, the standard form of the equation is \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \). Plugging the obtained values \(a=36\) and \(b=144\) into the equation returns: \( \frac{x^{2}}{36^{2}} - \frac{y^{2}}{144^{2}} = 1 \). This can be simplified further to \( \frac{x^{2}}{1296} - \frac{y^{2}}{20736} = 1 \).

Key Concepts

Foci of a Hyperbola

In the context of hyperbolas, the foci are two crucial points that help to define the shape and orientation of the curve. The foci are designated as fixed points, and for any given point on the hyperbola, the absolute difference in distances to these two foci remains constant.
This property gives the hyperbola its distinctive open curve shape. In our example, the foci are located at the points \((0,0)\) and \((300,0)\), indicating a horizontal orientation of the hyperbola.

• The center of the hyperbola is midway between the foci, making it at the point \((150,0)\).
• The distance between each focus and the center is denoted as \(c\).
• In this case, \(c = 150\) units.

Understanding the distance between foci and their position aids in formulating the equation of the hyperbola and ensures accurate graphing.

Distance Formula in Hyperbolas

A hyperbola is defined by the property that the absolute difference in distances from any point on the curve to the two foci is a constant value. This characteristic is at the heart of constructing hyperbolas.
For the problem at hand, all points on the hyperbola are 72 units closer to one focus than the other, meaning that this constant difference equals 72. In hyperbolas, this distance is referred to as \(2a\), where \(a\) is the semi-major axis, the distance from the center to a vertex on the hyperbola.

• Since \(2a = 72\), we find that \(a = 36\) units.

This value of \(a\) is essential in writing the standard form equation for the hyperbola.

Standard Form of a Hyperbola

The standard form of a hyperbola's equation provides a systematic way to express the curve in relation to its axes and important physical dimensions such as the distances \(a\) and \(b\). For a hyperbola aligned along the x-axis (a horizontal hyperbola), the standard equation is:
\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]
Here, \(a\) and \(b\) represent half the lengths of the transverse and conjugate axes, respectively. Given the previous calculations:

• \(a = 36\) gives \(a^2 = 1296\).
• \(b\) must be determined using the formula \(c^2 = a^2 + b^2\).
• Since \(c = 150\), substituting the values gives:
  \[b = \sqrt{c^2 - a^2} = \sqrt{150^2 - 36^2} = 144\]
• Thus \(b^2 = 20736\).

Substituting \(a^2\) and \(b^2\) into the equation results in the hyperbola:
\[\frac{x^2}{1296} - \frac{y^2}{20736} = 1\]
This standard form efficiently describes the hyperbola's shape and location, facilitating further manipulations and analyses.