Problem 72

Question

Find an equation for each hyperbola. $$\text { Foci }(-3 \sqrt{5}, 0) \text { and }(3 \sqrt{5}, 0) ; \text { asymptotes } y=\pm 2 x$$

Step-by-Step Solution

Verified

Answer

The hyperbola's equation is $\frac{x^2}{9} - \frac{y^2}{36} = 1$.

1Step 1: Identify Hyperbola Parameters

Given the foci at (-3\sqrt{5}, 0) and (3\sqrt{5}, 0), the hyperbola is centered at the origin (0, 0), and its major axis is horizontal. The distance from the center to each focus is $c = 3\sqrt{5}$.

2Step 2: Calculate $c^2$

We compute $c^2$ using the distance to the foci. Since $ c = 3\sqrt{5} $, we have $ c^2 = (3\sqrt{5})^2 = 45 $.

3Step 3: Find the Relationship of Asymptotes to $a$ and $b$

The asymptotes of the hyperbola are given by $y=\pm 2x$, which compares to the standard form of asymptotes $y = \pm \frac{b}{a}x$. Therefore, $\frac{b}{a}=2$.

4Step 4: Determine $a^2$ and $b^2$

Using the relationship $b^2 = 4a^2$ from $\frac{b}{a}=2$ and the equation $c^2 = a^2 + b^2$, we substitute to get $45 = a^2 + 4a^2=5a^2$. Solving for $a^2$, we find $a^2 = 9$. Hence, $b^2 = 36$.

5Step 5: Formulate the Hyperbola Equation

The standard form of the equation for a hyperbola with a horizontal transverse axis is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. Plugging in $a^2 = 9$ and $b^2 = 36$, the equation becomes $\frac{x^2}{9} - \frac{y^2}{36} = 1$.

Key Concepts

FociAsymptotesDistance FormulaEquation of Hyperbola

Foci

The foci of a hyperbola are two fixed points located along the hyperbola's major axis. They play a crucial role in defining the shape of the hyperbola. In our given problem, the foci are

located at
- $(-3\sqrt{5}, 0)$
- $(3\sqrt{5}, 0)$

The foci help to establish the center of the hyperbola, which is the midpoint between them. In this case, the center is

(0, 0).

The distance from the center to each focus is denoted as $c$, which indicates the focal distance. Here, $c$ is calculated as

$3\sqrt{5}$.

This distance is essential for forming the hyperbola's equation, as it affects both the size and orientation.

Asymptotes

Asymptotes of a hyperbola are the straight lines that the curve approaches but never touches. They provide a visual guide to the 'flattening' shape of the hyperbola. The asymptotes for our specific hyperbola are

$y = \pm 2x$.

These lines form a cross pattern through the center, indicating the steepness of the hyperbola's branches. The equation of the asymptotes can be compared to the standard hyperbola form

$y = \pm \frac{b}{a}x$.

This reveals the relationship between $a$ and $b$. In this problem, the equation yields a ratio of

$\frac{b}{a} = 2$,

which is fundamental for determining the relation between the transverse and conjugate axes lengths, helping to understand how the hyperbola stretches.

Distance Formula

To calculate the position of the foci in relation to the center, the distance formula is incredibly useful. It establishes how far apart key components of the hyperbola are from each other, enabling precise determination of the hyperbola's size. The distance formula is expressed as:

\[ d = c = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

For our exercise's given foci:

$(-3\sqrt{5}, 0)$ and $(3\sqrt{5}, 0)$,

the calculation becomes straightforward since both foci lie on the x-axis. The distance $c$ is therefore directly linked to the x-coordinates, giving us

$c = 3\sqrt{5}$.

This information is used to find $c^2 = 45$. Knowing $c$ and its square derivative plays an integral part in setting up the hyperbola's equation and understanding its structure.

Equation of Hyperbola

The equation of a hyperbola is the mathematical expression that describes its shape and position. For a hyperbola centered at the origin with a horizontal major axis, the standard form of its equation is: