The center of a hyperbola is located at the origin. One focus is located at [tex]$(-50,0)$[/tex] and its associated directrix is represented by the line [tex]$x=\frac{2,304}{50}$[/tex].

\begin{tabular}{|l|l|}
\hline
Vertices: [tex]$(-a, 0),(a, 0)$[/tex] & Vertices: [tex]$(0,-a),(0, a)$[/tex] \\
Foci: [tex]$(-c, 0),(c, 0)$[/tex] & Foci: [tex]$(0,-c),(0, c)$[/tex] \\
Asymptotes: [tex]$y= \pm \frac{b}{a} x$[/tex] & Asymptotes: [tex]$y= \pm \frac{a}{b} x$[/tex] \\
Directrices: [tex]$x= \pm \frac{a^2}{c}$[/tex] & Directrices: [tex]$y= \pm \frac{a^2}{c}$[/tex] \\
Standard Equation: [tex]$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$[/tex] & Standard Equation: [tex]$\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$[/tex] \\
\hline
\end{tabular}

What is the equation of the hyperbola?

[tex]$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$[/tex]



Answer :

To find the equation of the hyperbola with the given information, let's break down the steps required.

### Step 1: Identify the center and focus
The center of the hyperbola is given at the origin [tex]\((0, 0)\)[/tex], and one focus is located at [tex]\((-50, 0)\)[/tex]. This tells us that the hyperbola is oriented horizontally, as the focus is along the x-axis.

### Step 2: Determine the value of [tex]\(c\)[/tex]
Since the focus at [tex]\((-50, 0)\)[/tex] is 50 units away from the center, the value [tex]\(c\)[/tex] (the distance from the center to the focus) is:
[tex]\[ c = 50 \][/tex]

### Step 3: Determine the directrix and [tex]\(a^2/c\)[/tex]
The directrix is given by the line [tex]\(x = \frac{2304}{50}\)[/tex]. This tells us the value of [tex]\(\frac{a^2}{c}\)[/tex], where [tex]\(a\)[/tex] and [tex]\(c\)[/tex] are the parameters of the hyperbola.

So, we have:
[tex]\[ \frac{a^2}{c} = \frac{2304}{50} \][/tex]
[tex]\[ \frac{a^2}{50} = 46.08 \][/tex]

### Step 4: Calculate [tex]\(a^2\)[/tex]
To find [tex]\(a^2\)[/tex], multiply both sides of the equation by [tex]\(c\)[/tex]:
[tex]\[ a^2 = 46.08 \times 50 \][/tex]
[tex]\[ a^2 = 2304.0 \][/tex]

### Step 5: Calculate [tex]\(b^2\)[/tex]
For hyperbolas, the relationship between the distances [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] is given by:
[tex]\[ c^2 = a^2 + b^2 \][/tex]

Substitute the values for [tex]\(a^2\)[/tex] and [tex]\(c^2\)[/tex]:
[tex]\[ 50^2 = 2304.0 + b^2 \][/tex]
[tex]\[ 2500 = 2304.0 + b^2 \][/tex]
[tex]\[ b^2 = 2500 - 2304.0 \][/tex]
[tex]\[ b^2 = 196.0 \][/tex]

### Step 6: Write the standard form of the equation
For a horizontally oriented hyperbola centered at the origin, the standard form of the equation is:
[tex]\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \][/tex]

### Conclusion
We have found the values [tex]\(a^2 = 2304.0\)[/tex] and [tex]\(b^2 = 196.0\)[/tex]. Therefore, the equation of the hyperbola is:
[tex]\[ \frac{x^2}{2304.0} - \frac{y^2}{196.0} = 1 \][/tex]