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

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

Which are the equations of the directrices?



Answer :

Given that the hyperbola is centered at the origin and has vertical vertices and foci, the relevant information aligns with the second column in your table.

The vertex is at [tex]\((0,36)\)[/tex], so we have [tex]\(a = 36\)[/tex].

The focus is at [tex]\((0,39)\)[/tex], so we have [tex]\(c = 39\)[/tex].

For a hyperbola with vertical transverse axis, the equations of the directrices are given by:
[tex]\[ y = \pm \frac{a^2}{c} \][/tex]

Here's the detailed step-by-step solution to determine the equations of the directrices:

1. Calculate [tex]\(a^2\)[/tex]:
[tex]\[a^2 = 36^2 = 1296\][/tex]

2. Calculate [tex]\( \frac{a^2}{c} \)[/tex]:
[tex]\[ \frac{a^2}{c} = \frac{1296}{39} \approx 33.23076923076923\][/tex]

3. Write the equations of the directrices:
[tex]\[y = \pm \frac{a^2}{c}\][/tex]
Which gives us the two equations:
[tex]\[ y = 33.23076923076923 \][/tex]
[tex]\[ y = -33.23076923076923 \][/tex]

Therefore, the equations of the directrices are [tex]\( y = 33.23076923076923 \)[/tex] and [tex]\( y = -33.23076923076923 \)[/tex].