To solve this problem, we need to find the value of [tex]\( b \)[/tex] to correctly complete the equation for the hyperbola. Let's walk through the steps one by one:
1. Given Information:
- The standard form of the hyperbola centered at the origin:
[tex]\[
\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
\][/tex]
- The value of [tex]\( a \)[/tex] is given as [tex]\( 24 \)[/tex].
- The directrix is given as [tex]\( x = \frac{575}{25} = 23 \)[/tex].
2. Directrix Relationship:
- For a hyperbola, the directrix [tex]\( x = \frac{a^2}{c} \)[/tex] can be used to find [tex]\( c \)[/tex].
Using the given directrix:
[tex]\[
x = \frac{a^2}{c} \quad \Rightarrow \quad 23 = \frac{24^2}{c}
\][/tex]
Solving for [tex]\( c \)[/tex]:
[tex]\[
c = \frac{576}{23} \approx 25.043
\][/tex]
3. Relationship Between [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
- For a hyperbola, the relationship is [tex]\( c^2 = a^2 + b^2 \)[/tex].
- We already have [tex]\( a = 24 \)[/tex] and [tex]\( c \approx 25.043 \)[/tex].
Finding [tex]\( b^2 \)[/tex]:
[tex]\[
c^2 = a^2 + b^2 \quad \Rightarrow \quad (25.043)^2 = 24^2 + b^2
\][/tex]
[tex]\[
627.173 = 576 + b^2
\][/tex]
Solving for [tex]\( b^2 \)[/tex]:
[tex]\[
b^2 = 627.173 - 576 \approx 51.174
\][/tex]
Thus:
[tex]\[
b \approx \sqrt{51.174} \approx 7.154
\][/tex]
4. Completing the Equation:
- The hyperbola has the form [tex]\( \frac{x^2}{24^2} - \frac{y^2}{b^2} = 1 \)[/tex].
- The value of [tex]\( b \approx 7.154 \)[/tex] completes the blank space.
So, the complete equation is:
[tex]\[
\frac{x^2}{24^2} - \frac{y^2}{7.154^2} = 1
\][/tex]
To summarize, the positive value [tex]\( \boxed{7.154} \)[/tex] correctly fills in the blank in the hyperbola's equation.
