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A hyperbola centered at the origin has a vertex at [tex]$(0,36)$[/tex] and a focus at [tex]$(0,39)$[/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}

Which are the equations of the directrices?

[tex]$x = \pm \frac{12}{13}$[/tex]



Answer :

GinnyAnswer
To determine the equations of the directrices for a hyperbola centered at the origin and with a vertical orientation, let's analyze the given information and apply the corresponding formula.

### Key Information:
- Vertex: [tex]\( (0, 36) \)[/tex]
- This tells us that [tex]\( a = 36 \)[/tex], where [tex]\( a \)[/tex] is the distance from the center to each vertex.
- Focus: [tex]\( (0, 39) \)[/tex]
- This tells us that [tex]\( c = 39 \)[/tex], where [tex]\( c \)[/tex] is the distance from the center to each focus.

### Formula for Directrices:
Since the hyperbola is vertically oriented, the equations of the directrices are given by:
[tex]\[ y = \pm \frac{a^2}{c} \][/tex]

### Calculation:
1. Square of [tex]\( a \)[/tex]:
[tex]\[ a^2 = 36^2 = 1296 \][/tex]

2. Directrices' position:
[tex]\[ \frac{a^2}{c} = \frac{1296}{39} = 33.23076923076923 \][/tex]

### Equations of the Directrices:
Thus, the equations of the directrices are:
[tex]\[ y = 33.23076923076923 \][/tex]
[tex]\[ y = -33.23076923076923 \][/tex]

Therefore, the directrices of the hyperbola centered at the origin with a vertical orientation are:
[tex]\[ y = 33.23076923076923 \][/tex]
[tex]\[ y = -33.23076923076923 \][/tex]

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