To determine the equation of the given hyperbola, we need to identify the appropriate values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
First, let's denote:
- Vertices at [tex]\( (0, 10) \)[/tex] and [tex]\( (0, -10) \)[/tex]
- Foci at [tex]\( (0, 26) \)[/tex] and [tex]\( (0, -26) \)[/tex]
From this information, we can deduce:
1. Value of [tex]\(a\)[/tex]:
- The distance from the center (which is at the origin [tex]\((0,0)\)[/tex]) to each vertex is 10 units.
- So, [tex]\( a = 10 \)[/tex].
2. Value of [tex]\(c\)[/tex]:
- The distance from the center to each focus is 26 units.
- So, [tex]\( c = 26 \)[/tex].
3. Relationship between [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
- Hyperbolas follow the relationship [tex]\( c^2 = a^2 + b^2 \)[/tex].
Given values:
[tex]\[ c = 26 \][/tex]
[tex]\[ a = 10 \][/tex]
Calculate [tex]\(a^2\)[/tex]:
[tex]\[ a^2 = 10^2 = 100 \][/tex]
Calculate [tex]\(c^2\)[/tex]:
[tex]\[ c^2 = 26^2 = 676 \][/tex]
Now, using the relationship [tex]\( c^2 = a^2 + b^2 \)[/tex]:
[tex]\[ 676 = 100 + b^2 \][/tex]
Solve for [tex]\(b^2\)[/tex]:
[tex]\[ b^2 = 676 - 100 \][/tex]
[tex]\[ b^2 = 576 \][/tex]
So, we have:
[tex]\[ a^2 = 100 \][/tex]
[tex]\[ b^2 = 576 \][/tex]
Now, substituting these values into the standard form for the equation of a hyperbola, we get:
[tex]\[ \frac{y^2}{100} - \frac{x^2}{576} = 1 \][/tex]
Therefore, the equation of the hyperbola is:
[tex]\[ \boxed{\frac{y^2}{100} - \frac{x^2}{576} = 1} \][/tex]
