To determine the correct equation representing the hyperbola with the given conditions, we need to consider the standard form of the hyperbola centered at [tex]\((0,0)\)[/tex], which has a vertex and a focus along the same axis.
Given that the vertex is at [tex]\((-48, 0)\)[/tex] and the focus is at [tex]\((50, 0)\)[/tex], we can infer the following:
1. Center: The center of the hyperbola is at [tex]\((0,0)\)[/tex].
2. Vertices: Since one vertex is [tex]\((-48, 0)\)[/tex], the distance from the center to the vertex is [tex]\(48\)[/tex]. This gives us [tex]\(a = 48\)[/tex].
3. Foci: The focus is at [tex]\((50, 0)\)[/tex], so the distance from the center to the focus is [tex]\(50\)[/tex]. This gives us [tex]\(c = 50\)[/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]
Given:
[tex]\[ a = 48 \][/tex]
[tex]\[ c = 50 \][/tex]
We can find [tex]\(b^2\)[/tex] as follows:
[tex]\[ c^2 = a^2 + b^2 \][/tex]
[tex]\[ 50^2 = 48^2 + b^2 \][/tex]
[tex]\[ 2500 = 2304 + b^2 \][/tex]
[tex]\[ b^2 = 2500 - 2304 \][/tex]
[tex]\[ b^2 = 196 \][/tex]
[tex]\[ b = \sqrt{196}= 14 \][/tex]
The equation of the hyperbola depends on its orientation. Since the vertex and the focus lie on the x-axis, the hyperbola opens horizontally. The standard form for a horizontally opening hyperbola centered at [tex]\((0,0)\)[/tex] is:
[tex]\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \][/tex]
Substituting the values we found:
[tex]\[ a^2 = 48^2 = 2304 \][/tex]
[tex]\[ b^2 = 14^2 = 196 \][/tex]
Therefore, the equation of the hyperbola is:
[tex]\[ \frac{x^2}{48^2} - \frac{y^2}{14^2} = 1 \][/tex]
So, the correct answer is:
[tex]\[ \frac{x^2}{48^2} - \frac{y^2}{14^2} = 1 \][/tex]
