To determine the length of the major axis of an ellipse when given the eccentricity and the distance between the foci, we can use the following relationships:
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The distance between the foci \(2c\) is given as \(3,410,000\) km. Thus, we find \(c\): \[ c = \frac{3,410,000}{2} = 1,705,000 \text{ km} \]
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The relationship between the semi-major axis \(a\), eccentricity \(e\), and \(c\) is given by: \[ e = \frac{c}{a} \] Rearranging this gives: \[ a = \frac{c}{e} \]
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Given that \(e = 0.571\), we can now calculate \(a\): \[ a = \frac{1,705,000}{0.571} \approx 2,984,390.82 \text{ km} \]
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The length of the major axis \(2a\) can then be calculated as: \[ 2a \approx 2 \times 2,984,390.82 \approx 5,968,781.64 \text{ km} \]
Rounding this to the nearest whole number gives approximately \(5,970,000\) km.
Now, for the second question regarding the eccentricity of a perfect circle:
The eccentricity of a perfect circle is: \[ 0 \]
Thus, the answers are:
- Length of the major axis: 5,970,000 km
- Eccentricity of a perfect circle: 0