To find the eccentricity of an ellipse given the distance between its foci and the length of its minor axis, follow these steps:
1. Identify the given values:
- The distance between the foci is given as 6.
- The length of the minor axis is given as 8.
2. Determine the semi-minor axis (b):
- The semi-minor axis (b) is half of the minor axis.
[tex]\[
b = \frac{8}{2} = 4
\][/tex]
3. Determine the focal distance (c):
- The distance between the foci is twice the focal distance from the center (2c), so:
[tex]\[
c = \frac{6}{2} = 3
\][/tex]
4. Find the semi-major axis (a):
- Using the relationship between the semi-major axis, semi-minor axis, and the focal distance:
[tex]\[
a^2 = b^2 + c^2
\][/tex]
Plugging in the values for [tex]\(b\)[/tex] and [tex]\(c\)[/tex]:
[tex]\[
a^2 = 4^2 + 3^2 = 16 + 9 = 25
\][/tex]
Therefore:
[tex]\[
a = \sqrt{25} = 5
\][/tex]
5. Calculate the eccentricity (e):
- The eccentricity of an ellipse is given by the formula:
[tex]\[
e = \frac{c}{a}
\][/tex]
Substituting the values of [tex]\(c\)[/tex] and [tex]\(a\)[/tex]:
[tex]\[
e = \frac{3}{5} = 0.6
\][/tex]
So, the eccentricity of the ellipse is [tex]\(0.6\)[/tex].
