Question

what are the lengths of the major and minor axes of the ellipse ( 9x^{2}+4y^{2}-36 = 0 )?
write your answer in simplified, rationalized form.
length of major axis =
length of minor axis =

Explanation:

Step1: Rewrite the ellipse equation in standard form

The given equation is \(9x^{2}+4y^{2}-36 = 0\). Add \(36\) to both sides: \(9x^{2}+4y^{2}=36\). Divide each term by \(36\): \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\). For an ellipse of the form \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}} = 1\) (\(a>b>0\)), the length of the major axis is \(2a\) and the length of the minor axis is \(2b\). Here \(a^{2}=9\), so \(a = 3\); \(b^{2}=4\), so \(b = 2\).

Step2: Calculate the lengths of the axes

The length of the major axis is \(2a\). Substituting \(a = 3\), we get \(2\times3=6\).
The length of the minor axis is \(2b\). Substituting \(b = 2\), we get \(2\times2 = 4\).

Answer:

Length of major axis \(=6\)
Length of minor axis \(=4\)