To determine which equation represents the hyperbola with a center at [tex]\((0, 0)\)[/tex] and a vertex at [tex]\((-48, 0)\)[/tex], let's analyze the properties and standard form of a hyperbola.
A hyperbola oriented along the x-axis takes the form:
[tex]\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \][/tex]
where [tex]\(a\)[/tex] is the distance from the center to each vertex on the x-axis, and [tex]\(b\)[/tex] is the distance from the center to each co-vertex on the y-axis.
Given:
- The center is at [tex]\((0, 0)\)[/tex]
- The vertex is at [tex]\((-48, 0)\)[/tex]
The distance from the center to the vertex is 48, hence:
[tex]\[ a = 48 \][/tex]
The equation for this hyperbola should therefore be:
[tex]\[ \frac{x^2}{48^2} - \frac{y^2}{b^2} = 1 \][/tex]
We now need to find the corresponding equation from the given choices:
1. [tex]\(\frac{x^2}{50^2} - \frac{y^2}{14^2} = 1\)[/tex]
2. [tex]\(\frac{x^2}{48^2} - \frac{y^2}{14^2} = 1\)[/tex]
3. [tex]\(\frac{y^2}{50^2} - \frac{x^2}{14^2} = 1\)[/tex]
4. [tex]\(\frac{y^2}{48^2} - \frac{x^2}{14^2} = 1\)[/tex]
Based on the values we've identified:
- [tex]\(a = 48\)[/tex]
- [tex]\(b = 14\)[/tex] (given in the denominator for [tex]\(y^2\)[/tex])
Thus, the correct equation should include [tex]\(a^2 = 48^2\)[/tex] and [tex]\(b^2 = 14^2\)[/tex] in the forms of:
[tex]\[ \frac{x^2}{48^2} - \frac{y^2}{14^2} = 1 \][/tex]
Among the given options, the second equation matches this form.
Therefore, the correct equation representing the hyperbola is:
[tex]\[
\boxed{\frac{x^2}{48^2} - \frac{y^2}{14^2} = 1}
\][/tex]
