Problem 26 - Precalculus: Functions and Graphs

Question

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Foci \(F(\pm 7,0), \quad\) conjugate axis of length 8

Step-by-Step Solution

Answer

The hyperbola's equation is \(\frac{x^2}{33} - \frac{y^2}{16} = 1\).

Step 1: Identify the Standard Form of Hyperbola

For a hyperbola centered at the origin with the foci on the x-axis, the standard form of the equation is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), where \(c\) is the distance from the center to each focus, \(a\) is half the length of the transverse axis, and \(b\) is half the length of the conjugate axis.

Step 2: Determine Values for c and b

The foci are given as \((\pm 7, 0)\), meaning \(c = 7\). The length of the conjugate axis is 8, so \(b = \frac{8}{2} = 4\).

Step 3: Use Relationship to Find a

For hyperbolas, the relationship between the axes and the foci is \(c^2 = a^2 + b^2\). Substitute \(c = 7\) and \(b = 4\) into the equation to find \(a\). Therefore, \(49 = a^2 + 16\).

Step 4: Solve for a^2

Rearrange the equation found in Step 3 to find \(a^2\): \(a^2 = 49 - 16 = 33\).

Step 5: Write the Hyperbola Equation

Using the values found: \(a^2 = 33\) and \(b^2 = 16\), substitute into the standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) to get \(\frac{x^2}{33} - \frac{y^2}{16} = 1\).

Key Concepts

Standard Form of Hyperbola

The concept of the standard form of a hyperbola is fundamental in understanding its equation. A hyperbola is a type of conic section, appearing as a pair of mirrored curves. When its center is at the origin, and the foci lie on the x-axis, the standard form is given by:

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

Here, \(a\) represents half the length of the transverse axis, and \(b\) represents half the length of the conjugate axis. The values of \(a^2\) and \(b^2\) are essential in shaping the hyperbola and determining the orientation of its curves. Understanding the components \(a^2\) and \(b^2\) helps us establish the equation of the hyperbola accurately. This form highlights how hyperbolas differ from other conics, such as ellipses, by having components subtracted rather than added.

Foci of Hyperbola

The foci of a hyperbola are central to its definition and serve as key points of reference. They are two fixed points further apart than any part of the actual hyperbola. In our problem, the foci are given as \((\pm 7, 0)\), which means:

• The foci lie along the x-axis.
• The distance, \(c\), from the center to each focus is 7.

Foci are critical because the difference in distances from any point on the hyperbola to each focus is always the same. This property defines the shape's elongated paths. In the equation \(c^2 = a^2 + b^2\), \(c\) helps in calculating \(a\), enhancing the precise shape and orientation of the hyperbola.

Conjugate Axis

The conjugate axis of a hyperbola provides structural balance and perpendicularity to the transverse axis. This axis is distinct as it doesn't intercept the hyperbola but instead relates to its width. In the given exercise, the length of the conjugate axis is 8, which means:

• The full conjugate axis stretches 8 units long.
• The half-length \(b\) is therefore 4, as calculated by \(b = \frac{8}{2}\).

The value \(b\) is pivotal in the equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). It affects the opening of the hyperbola's curves. Additionally, knowing the conjugate axis length aids in determining the spatial distribution of the hyperbola's arms around the center.

Transverse Axis

The transverse axis in a hyperbola is the main axis that passes through the foci. It is paramount to the hyperbola's equation and determines the direction and opening of the curves. For a horizontal hyperbola focused along the x-axis, as in this exercise:

• \(a\) is half the length of the transverse axis.
• \(a^2\) is calculated using the equation \(c^2 = a^2 + b^2\).

The equation rearranges to solve for \(a^2\), providing \(a^2 = 33\). The transverse axis directly affects the hyperbola's stretch along the x-axis, marking the path and spread of its curves. It equates and balances with the conjugate axis to create a unique conic section, defined mathematically and visually by its distinct orientation and symmetry.