To determine the foci of the hyperbola given by the equation [tex]\(\frac{y^2}{5^2} - \frac{x^2}{9^2} = 1\)[/tex], we need to follow a detailed, step-by-step process.
1. Identify the standard form of the hyperbola equation:
The given equation is [tex]\(\frac{y^2}{5^2} - \frac{x^2}{9^2} = 1\)[/tex]. This is in the form [tex]\(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\)[/tex].
2. Determine [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
Here, [tex]\(a = 5\)[/tex] and [tex]\(b = 9\)[/tex].
3. Find [tex]\(c\)[/tex] using the relationship for hyperbolas:
For hyperbolas, the relationship between [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] is given by:
[tex]\[
c^2 = a^2 + b^2
\][/tex]
Substituting the known values:
[tex]\[
c^2 = 5^2 + 9^2
\][/tex]
[tex]\[
c^2 = 25 + 81
\][/tex]
[tex]\[
c^2 = 106
\][/tex]
Therefore:
[tex]\[
c = \sqrt{106}
\][/tex]
4. Determine the coordinates of the foci:
For a hyperbola of the form [tex]\(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\)[/tex], the foci are located at [tex]\((\pm c, 0)\)[/tex].
5. Substitute the value of [tex]\(c\)[/tex] into the foci coordinates:
The foci of the hyperbola are [tex]\((\pm \sqrt{106}, 0)\)[/tex].
Therefore, the correct answer is:
C. [tex]\(( \pm \sqrt{106}, 0)\)[/tex]
