Question

what is the eccentricity of the ellipse shown below?\\(
\frac{(x - 5)^2}{52} + \frac{(y + 1)^2}{64} = 1\\)
a. \\(\frac{2}{\sqrt{3}}\\)
b. \\(\frac{\sqrt{3}}{2}\\)
c. \\(\frac{\sqrt{3}}{4}\\)
d. \\(\sqrt{3}\\)

Explanation:

Step1: Identify \(a^2\) and \(b^2\)

For the ellipse \(\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1\) (\(a>b\)), from \(\frac{(x - 5)^2}{52}+\frac{(y + 1)^2}{64}=1\), we have \(a^2 = 64\), so \(a = 8\), and \(b^2=52\).

Step2: Use the formula \(c^2=a^2 - b^2\)

Calculate \(c^2\): \(c^2=64 - 52=12\), then \(c=\sqrt{12}=2\sqrt{3}\).

Step3: Use the eccentricity formula \(e=\frac{c}{a}\)

Substitute \(c = 2\sqrt{3}\) and \(a = 8\) into \(e=\frac{c}{a}\), we get \(e=\frac{2\sqrt{3}}{8}=\frac{\sqrt{3}}{4}\).

Answer:

C. \(\frac{\sqrt{3}}{4}\)