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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]
### 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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