Question

what is the length of the major axis of the ellipse $\frac{x^{2}}{19}+\frac{y^{2}}{100}=1$? write your answer in simplified, rationalized form.

Explanation:

Step1: Identify the form of the ellipse equation

The standard form of an ellipse equation is \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}} = 1\) (\(a>b>0\)), where the major - axis length is \(2a\).
For the given ellipse \(\frac{x^{2}}{19}+\frac{y^{2}}{100}=1\), we have \(a^{2}=100\) and \(b^{2}=19\).

Step2: Solve for \(a\)

Since \(a^{2}=100\), then \(a=\sqrt{100}=10\) (we take the positive value because \(a\) represents a length).

Step3: Calculate the length of the major axis

The length of the major axis of the ellipse \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\) is \(2a\).
Substituting \(a = 10\) into \(2a\), we get \(2\times10=20\).

Answer:

\(20\)