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]
