Problem 31 - Algebra 2

Question

What is an equation for the hyperbola centered at the origin with a vertical transverse axis of length 12 units and a conjugate axis of length 4 units?

Step-by-Step Solution

Answer

The equation of the hyperbola is \( \frac{y^2}{36} - \frac{x^2}{4} = 1 \).

Step 1: Understand the Equation of a Hyperbola

The general formula for a hyperbola centered at the origin with a vertical transverse axis is given by:\[\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\]where \(a\) represents the semi-transverse axis and \(b\) represents the semi-conjugate axis.

Step 2: Identify the Lengths of Semi-Axes

The problem states the length of the transverse axis is 12 units and length of the conjugate axis as 4 units. The semi-transverse axis \(a\) is half of the transverse axis, so \(a = \frac{12}{2} = 6\).Likewise, the semi-conjugate axis \(b\) is half of the conjugate axis, so \(b = \frac{4}{2} = 2\).

Step 3: Substitute and Simplify

Substitute \(a\) and \(b\) into the formula:\[\frac{y^2}{6^2} - \frac{x^2}{2^2} = 1\]This simplifies to:\[\frac{y^2}{36} - \frac{x^2}{4} = 1\]

Key Concepts

Vertical Transverse Axis

In a hyperbola, the transverse axis is the line segment that passes through the center and connects the two vertices of the hyperbola. When this axis is vertical, it implies that the curve opens upwards and downwards rather than sideways. This orientation changes the standard equation of the hyperbola. To accommodate this vertical orientation, the formula changes from the horizontal form to:

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

Here, the larger value is associated with \(y^2\), indicating that the primary stretching occurs vertically along the \(y\)-axis. This distinguishes it from a hyperbola with a horizontal transverse axis where \(x^2\) would hold the larger divisor.

Semi-Transverse Axis

The semi-transverse axis, commonly referred to simply as "a," is a crucial part of defining the hyperbola's size and orientation. It signifies half of the full length of the transverse axis. In mathematical terms:

• \(a = \frac{L}{2}\)

where \(L\) is the full length of the transverse axis. For a hyperbola with a given transverse axis length of 12 units, the semi-transverse axis would be 6 units. This value is squared and placed under \(y^2\) in the equation when dealing with a vertical transverse axis.

Semi-Conjugate Axis

The semi-conjugate axis is represented by "b" and is half of the total length of the conjugate axis. It plays an essential role in describing the "width" of the hyperbola. Similar to the semi-transverse axis, it is calculated as:

• \(b = \frac{C}{2}\)

where \(C\) is the full length of the conjugate axis. For our example with a conjugate axis of 4 units, the semi-conjugate axis would be 2 units. This value is crucial for constructing the hyperbola’s equation, as it is squared and placed under \(x^2\) in the formula for a vertical transverse axis.

Centered at Origin

Hyperbolas can be centered at various points in the coordinate plane, but when centered at the origin, the equations simplify significantly. The standard format for a hyperbola centered at the origin with a vertical transverse axis is:

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

This format identifies the center point \((0,0)\), making calculations straightforward. The symmetry around both axes is clear, making it a favored representation for many problems. A hyperbola centered at the origin will always maintain this symmetric equation relationship regardless of the axis orientation.