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The equation [tex]\frac{x^2}{24^2}-\frac{y^2}{b^2}=1[/tex] represents a hyperbola centered at the origin with a directrix of [tex]x=\frac{576}{26}[/tex].

\begin{tabular}{|l|l|}
\hline
Vertices: [tex]$(-a, 0),(a, 0)$[/tex] & Vertices: [tex]$(0,-a),(0, a)$[/tex] \\
Foci: [tex]$(-c, 0),(c, 0)$[/tex] & Foci: [tex]$(0,-c),(0, c)$[/tex] \\
Asymptotes: [tex]$y= \pm \frac{b}{a} x$[/tex] & Asymptotes: [tex]$y= \pm \frac{a}{b} x$[/tex] \\
Directrices: [tex]$x= \pm \frac{a^2}{c}$[/tex] & Directrices: [tex]$y= \pm \frac{a^2}{c}$[/tex] \\
Standard Equation: [tex]$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$[/tex] & Standard Equation: [tex]$\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$[/tex] \\
\hline
\end{tabular}

What is the value of [tex]b[/tex]?

A. 10

B. 16

C. 20

D. 26



Answer :

GinnyAnswer
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].

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