Problem 50 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the standard form of the equation of the hyperbola with the given characteristics. Foci: (1,2),(1,6) asymptotes: \(y=2+2 x, y=6-2 x\)

Step-by-Step Solution

Answer

The equation of the hyperbola is \( (y-4)^2/4 - (x-1)^2 = 1 \)

Step 1: Calculate distance 'c'

We calculate the distance between a focus and the center, known as 'c'. Given foci at (1,2) and (1,6), we find that 'c' is 2.

Step 2: Determine 'a' and 'b' from the asymptotes

We see that the slopes in the equations of the asymptotes are ±2. We have \(y-k = ±(x-h)\), and the center of the hyperbola is the average of the y-coordinates of the foci, which is 4, and its x-coordinate is 1. This means that \(a = 2\) and \(b = 1\).

Step 3: Substitute values into the formula

We substitute the values we got in Step 1 and Step 2 into the equation for a vertical hyperbola and get \( (y-4)^2/2^2 - (x-1)^2/1^2 = 1 \), which simplifies to \( (y-4)^2/4 - (x-1)^2 = 1 \)

Key Concepts

Foci of a Hyperbola

A hyperbola is a fascinating structure in mathematics, formed by all points for which the difference in distances to two fixed points, called the foci, is constant. The foci of a hyperbola play a crucial role in determining the shape and position of the hyperbola. For any hyperbola, there are always exactly two foci.

In the given exercise, the foci are at the coordinates (1,2) and (1,6). These points help in identifying crucial features like the center and the orientation of the hyperbola.

• Center: The center is located halfway between the foci. We find it by averaging the coordinates of the foci. For this exercise, the center is (1,4).
• Distance 'c': The distance from the center to one of the foci is denoted by 'c'. Here, since the y-values differ by 4 units, and they are spread symmetrically around the center's y-coordinate 4, 'c' equals 2.

Understanding the properties of the foci of a hyperbola is key to solving many problems related to its equations and graphs.

Asymptotes of a Hyperbola

Asymptotes are lines that the branches of a hyperbola approach but never touch. They are helpful in sketching the hyperbola as they provide the direction of the curve. In the standard form of a hyperbola, the slopes of the asymptotes give insights into the relationship between 'a' and 'b', which are the measures of the transverse and conjugate axes, respectively.

The given asymptotes in this exercise are represented by the equations:

• Positive Slope: \( y = 2 + 2x \)
• Negative Slope: \( y = 6 - 2x \)

The slopes are ±2, meaning for every 1 unit along the x-axis, y changes by 2 units. This information indicates that the division of lengths in the hyperbola (a to b) is 2:1. Hence, 'b' is smaller than 'a'. With the center known, the asymptotes provide a frame around which the hyperbola is shaped.

Standard Form of a Hyperbola Equation

The standard form of the equation of a hyperbola offers a systematic representation for identifying all its elements, such as the center, axes, and orientation. Depending on the orientation (vertical or horizontal), the form will differ slightly.

For this exercise, we are dealing with a vertically oriented hyperbola. Thus, the standard form of its equation is \[\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\]where:

• (h, k): Center coordinates, for this hyperbola, it's (1,4).
• a: Distance along the y-direction from the center to the vertex, here 2.
• b: Distance along the x-direction perpendicular to 'a', here 1.

Substituting these into the standard formula gives \( \frac{(y-4)^2}{4} - \frac{(x-1)^2}{1} = 1 \). This equation fully characterizes the shape and position of the hyperbola.

Distance Between Points

Understanding how to calculate the distance between points is fundamental in geometry and helps determine properties like spacing between foci or vertices. The distance formula computes the distance between two points in a plane and is given by:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

In our exercise, we use this concept to calculate 'c', the distance from the center to a focus. This helps in forming the hyperbola's equation. Given foci (1,2) and (1,6), these points only differ in the y-coordinate. Hence, using the distance formula:

• Subtract the y-values: \(6 - 2 = 4\)
• The x-values are the same, so \(x_2 - x_1 = 0\)
• The actual distance thus simplifies to 4, but since the center-to-focus, 'c', is essentially half (symmetrically divided by the center (1,4)), hence 'c' is 2.

Proper determination of these distances is a step toward correctly understanding the hyperbola's structure.