Step 1: Understanding the Question:
This question is based on the properties of an ellipse, which is a key topic in coordinate geometry.
We are given that the major axis of an ellipse is three times the length of its minor axis, and we need to calculate its eccentricity.
Step 2: Key Formulas and Approach:
Ellipse Axes: For a standard ellipse, the length of the major axis is $2a$ and the minor axis is $2b$ (assuming $a>b$).
Eccentricity Formula:
\[
e = \sqrt{1 - \frac{b^2}{a^2}}
\]
Our approach is to establish a relationship between $a$ and $b$ from the given axis ratio, find the value of $\frac{b^2}{a^2}$, and substitute it into the eccentricity formula.
Step 3: Detailed Explanation:
Establish the Axis Relationship:
According to the problem statement:
\[
\text{Major Axis} = 3 \times \text{Minor Axis}
\]
\[
2a = 3 \times (2b) \quad \implies \quad a = 3b
\]
Find the Ratio of the Semi-axes:
We can express $\frac{b}{a}$ as:
\[
\frac{b}{a} = \frac{1}{3}
\]
Squaring both sides of the ratio:
\[
\frac{b^2}{a^2} = \frac{1}{9}
\]
Calculate the Eccentricity ($e$):
Substitute the value of $\frac{b^2}{a^2}$ into the eccentricity formula:
\[
e = \sqrt{1 - \frac{1}{9}}
\]
\[
e = \sqrt{\frac{8}{9}} = \frac{\sqrt{8}}{3}
\]
Simplifying the numerator:
\[
\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}
\]
Thus, the eccentricity is:
\[
e = \frac{2\sqrt{2}}{3}
\]
Step 4: Final Answer:
The eccentricity of the ellipse is $\frac{2\sqrt{2}}{3}$, which corresponds to Option (C).