To solve for [tex]\( b \)[/tex] in the equation of the hyperbola [tex]\(\frac{x^2}{24^2} - \frac{y^2}{b^2} = 1\)[/tex] with a given directrix [tex]\(x = \frac{576}{26}\)[/tex], we follow these steps:
1. Identify the given parameters:
- [tex]\( a = 24 \)[/tex] (since [tex]\( \frac{x^2}{24^2} \)[/tex] gives us that [tex]\( a \)[/tex] is 24)
- Directrix is [tex]\( x = \frac{576}{26} \)[/tex]
2. Recall the formula for the directrix:
The directrix of a hyperbola [tex]\( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)[/tex] is given by [tex]\( x = \pm \frac{a^2}{c} \)[/tex].
3. Calculate [tex]\( c \)[/tex] using the directrix information:
[tex]\[
\frac{a^2}{c} = \frac{576}{26}
\][/tex]
Therefore,
[tex]\[
c = \frac{a^2}{\frac{576}{26}} = \frac{24^2 \times 26}{576}
\][/tex]
Simplify the calculation:
[tex]\[
a^2 = 576
\][/tex]
[tex]\[
c = \frac{576 \times 26}{576} = 26
\][/tex]
4. Use the relationship between [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] in a hyperbola:
For a hyperbola, [tex]\( c^2 = a^2 + b^2 \)[/tex].
5. Plug in the values of [tex]\( a \)[/tex] and [tex]\( c \)[/tex]:
[tex]\[
c = 26
\][/tex]
[tex]\[
c^2 = a^2 + b^2 \implies 26^2 = 24^2 + b^2
\][/tex]
[tex]\[
676 = 576 + b^2
\][/tex]
[tex]\[
b^2 = 676 - 576
\][/tex]
[tex]\[
b^2 = 100
\][/tex]
[tex]\[
b = \sqrt{100}
\][/tex]
[tex]\[
b = 10
\][/tex]
6. Answer:
The value of [tex]\( b \)[/tex] is [tex]\( 10 \)[/tex].
