Final answer:
The eccentricity of an ellipse is the ratio of the distance between the foci and the length of the major axis. To find it, calculate the distances from the given point to the foci, sum them to find the major axis (2a), and then divide the distance between the foci by the major axis length (e = c/a). Compare the result to the options provided to find the correct answer.
Step-by-step explanation:
To find the eccentricity of the ellipse, we first need to understand a few properties of an ellipse. The eccentricity (e) is a measure of how much the ellipse deviates from being circular, and it is defined as the distance between the foci (2c) divided by the length of the major axis (2a).
An ellipse has two foci, and the sum of the distances from any point on the ellipse to the foci is constant. This constant is the length of the major axis (2a). Given that the ellipse passes through the point (2√13, 4√2) and has its foci at (-4, 1) and (4, 1), we can first find the major axis length by using the distance formula to calculate the distances from the point to each focus and then summing them up.
The distance between the foci, which are aligned along the x-axis, is 8 units (from -4 to 4). Hence, 2c = 8, and c = 4. Now, using the fact that the sum of the distances from the point to the foci equals 2a, we have:
Distance to first focus: √[(2√13 - (-4))² + (4√2 - 1)²]
Distance to second focus: √[(2√13 - (4))² + (4√2 - 1)²]
Sum of distances = 2a
Calculating the distances and summing them will yield 2a, which we then use to calculate the eccentricity as e = c/a.
Once you find a, you plug it into the eccentricity formula, compare the calculated value to the options provided in the question, and select the correct answer.