To solve for the positive value that correctly fills in the blank in the equation [tex]\(\frac{x^2}{24^2} - \frac{y^2}{[\square]^2} = 1\)[/tex], follow these steps:
1. Identify the given parameters:
- The hyperbola equation is [tex]\(\frac{x^2}{24^2} - \frac{y^2}{[\square]^2} = 1\)[/tex], which indicates [tex]\(a = 24\)[/tex] and we need to find [tex]\(b\)[/tex].
- The directrix is given by the equation [tex]\(x = \frac{576}{26}\)[/tex].
2. Compute the directrix value:
- Calculate the directrix value: [tex]\(x = \frac{576}{26}\)[/tex]:
[tex]\[
x = \frac{576}{26} = 22.153846153846153
\][/tex]
3. Compute the value of [tex]\(c\)[/tex]:
- The relationship between the directrix and the hyperbola parameters is given by the formula [tex]\(x = \frac{a^2}{c}\)[/tex], where [tex]\(c\)[/tex] is the distance from the center to the foci:
[tex]\[
22.153846153846153 = \frac{24^2}{c}
\][/tex]
- Solve for [tex]\(c\)[/tex]:
[tex]\[
c = \frac{24^2}{22.153846153846153}
\][/tex]
- The value obtained for [tex]\(c\)[/tex] is:
[tex]\[
c = 26.0
\][/tex]
4. Use the relationship between [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
- In a hyperbola, the relationship between [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] is given by:
[tex]\[
c^2 = a^2 + b^2
\][/tex]
- Substitute [tex]\(a = 24\)[/tex] and [tex]\(c = 26.0\)[/tex]:
[tex]\[
26^2 = 24^2 + b^2
\][/tex]
- Simplify and solve for [tex]\(b^2\)[/tex]:
[tex]\[
676 = 576 + b^2
\][/tex]
[tex]\[
b^2 = 676 - 576
\][/tex]
[tex]\[
b^2 = 100
\][/tex]
5. Find [tex]\(b\)[/tex]:
- Calculate [tex]\(b\)[/tex]:
[tex]\[
b = \sqrt{100} = 10.0
\][/tex]
Therefore, the positive value that correctly fills in the blank in the equation [tex]\(\frac{x^2}{24^2} - \frac{y^2}{[\square]^2} = 1\)[/tex] is [tex]\(\boxed{10}\)[/tex].
