If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the eccentricity of the hyperbola is:

Let 2b be the length of the conjugate axis and 2ae be the distance between the foci.
Given that the length of the conjugate axis is 5, we have 2b = 5, so b = 5/2 = 2.5.
The distance between the foci is 13, so 2ae = 13.
For a hyperbola, the relationship between a, b, and e (eccentricity) is given by b² = a²(e² - 1).
We have 2ae = 13, so ae = 13/2 = 6.5.
Also, we have b = 2.5, so b² = (2.5)² = 6.25.
Substituting into b² = a²(e² - 1), we get:
6.25 = a²(e² - 1)
Since ae = 6.5, we have a = 6.5/e.
Substituting this into the equation above:
6.25 = (6.5/e)²(e² - 1)
6.25 = (42.25/e²)(e² - 1)
6.25e² = 42.25e² - 42.25
36e² = 42.25
e² = 42.25/36 = 1.1736
e = √1.1736 ≈ 1.0833
However, this value of e is incorrect as it should be greater than 1. Let's reconsider the approach.
We have 2b = 5, so b = 2.5. We also have 2ae = 13, so ae = 6.5.
The relationship between a, b, and e is a² + b² = (ae)².
Substituting the values, we get a² + (2.5)² = (6.5)²
a² + 6.25 = 42.25
a² = 36
a = 6
Now, we have ae = 6.5, so 6e = 6.5, which gives e = 6.5/6 = 13/12.
Therefore, the eccentricity of the hyperbola is 13/12.