Problem 36 - Precalculus: Functions and Graphs

Question

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Horizontal transverse axis of length \(6,\) conjugate axis of length 2

Step-by-Step Solution

Answer

The equation is \( \frac{x^2}{9} - \frac{y^2}{1} = 1 \).

Step 1: Understanding the Standard Form of a Hyperbola

For a hyperbola centered at the origin with a horizontal transverse axis, the equation is of the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). In this case, \( a \) represents half the length of the transverse axis and \( b \) represents half the length of the conjugate axis.

Step 2: Calculating Half the Length of the Transverse and Conjugate Axes

The length of the transverse axis is given as 6, so \( a = \frac{6}{2} = 3 \). The length of the conjugate axis is given as 2, so \( b = \frac{2}{2} = 1 \).

Step 3: Substituting Values into the Standard Form

Now, substitute \( a = 3 \) and \( b = 1 \) into the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) to get the equation of the hyperbola: \( \frac{x^2}{3^2} - \frac{y^2}{1^2} = 1 \).

Step 4: Simplifying the Equation

Simplify the equation by calculating \( a^2 \) and \( b^2 \). Thus, \( 3^2 = 9 \) and \( 1^2 = 1 \). Therefore, the equation of the hyperbola is \( \frac{x^2}{9} - \frac{y^2}{1} = 1 \).

Key Concepts

Horizontal Transverse Axis

In the context of hyperbolas, the transverse axis plays a crucial role. It is the axis along which the two branches of the hyperbola open. When the transverse axis is horizontal, it implies that these branches expand left and right. For instance, in our problem, the length of the horizontal transverse axis is given as 6. This means that from the center of the hyperbola, at the origin, the axis extends 3 units left and 3 units right. This is because the length extends equally on both sides of the center. To plug this into the equation of the hyperbola, we use the standard form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). In this form, \( a \), which represents half the length of the transverse axis, is equal to 3. Hence, \( a^2 = 9 \), contributing to the denominator under \( x^2 \) in the equation representing the slope of the branches.

Conjugate Axis

The conjugate axis of a hyperbola complements the transverse axis and is perpendicular to it. It reflects the symmetry structure and gives spaces between the branches above and below the center point. In a standard hyperbola, the length of the conjugate axis is influenced by the value of \( b \). For this exercise, the conjugate axis has a length of 2, which suggests that it extends 1 unit above and 1 unit below the center at the origin. Thus, \( b = 1 \) and \( b^2 = 1 \). This is incorporated into the equation in the denominator under \( y^2 \) for \( \frac{y^2}{b^2} \).It is important to understand how both axes together define the 'box' which helps in sketching the hyperbola. This 'box' further determines the slopes of the asymptotes, intrinsic to understanding hyperbolas' behavior further from their center.

Center at the Origin

When the hyperbola's center is at the origin (0,0), it simplifies how we perceive its equation and graph. The origin as the center means that both axes of the hyperbola symmetrically stretch from this central point, without causing translation in any specific direction. For the hyperbola equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the center at the origin ensures that none of the terms involve an extra \( h \) or \( k \) to indicate shiftings, such as \( (x-h) \) or \( (y-k) \). It allows focusing mainly on the scaling and ratios dictated by the axes' lengths. Thus, in its cleanest form, this hyperbolic equation represents the symmetrical behavior of the two branches opening away from the origin along the defined axis parameters.