Question

what is the length of the major axis of the ellipse \\( \frac{(x - 5)^{2}}{9}+\frac{y^{2}}{16}=1 \\)? write your answer in simplified, rationalized form.

Explanation:

Step1: Identify the form of the ellipse equation

The standard form of an ellipse is \(\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1\) (\(a>b\)) where the major axis length is \(2a\).
Given \(\frac{(x - 5)^2}{9}+\frac{y^2}{16}=1\), we have \(a^{2}=16\), \(b^{2}=9\).

Step2: Calculate the value of \(a\)

Since \(a^{2}=16\), then \(a = 4\) (because \(a>0\)).

Step3: Find the length of the major axis

The length of the major axis is \(2a\). Substituting \(a = 4\), we get \(2\times4=8\).

Answer:

\(8\)