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Normalization is a crucial step in simplifying the equation of an ellipse. It helps in representing the ellipse in a standard form that is easier to analyze. To normalize an ellipse equation, you divide each term by a constant to make the equation equal to 1. In the given problem, the original equation is

• \(9x^{2} + 4y^{2} = 36\)

To normalize it, you divide every term by 36, which is the right side of the equation. This changes the equation to:

• \(\frac{x^{2}}{4} + \frac{y^{2}}{9} = 1\)

This form is important because it allows us to clearly see the relationship between the variables and helps identify key features of the ellipse, like its axes.

In the context of ellipses, the semi-major axis and the semi-minor axis are key components defining the ellipse's shape and orientation. Once the ellipse equation is normalized as

• \(\frac{x^{2}}{4} + \frac{y^{2}}{9} = 1\)

we can identify the denominators of the terms to find the lengths of these axes. The semi-major axis corresponds to the largest denominator in the normalized equation, which is 9 in this example. Therefore, the semi-major axis lies along the y-direction, and its length is the square root of 9, hence 3.On the other hand, the semi-minor axis corresponds to the smaller denominator, which is 4. This means it's along the x-direction, with its length being the square root of 4, giving us 2.These axes are crucial because they tell us the extent of the ellipse in both directions, which is essential for graphing and understanding its proportions.

The general equation of an ellipse, when centered at the origin (0,0), takes a specific form in its normalized state:

• \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)

Here, \(a^2\) and \(b^2\) are the denominators showing the squared values of the semi-major and semi-minor axes, respectively. An ellipse centered at the origin is easier to work with mathematically, as you don't need to adjust for any horizontal or vertical shifts.By analyzing the normalized equation from the problem:

• \(\frac{x^2}{4} + \frac{y^2}{9} = 1\)

we can see that \(a^2 = 4\) and \(b^2 = 9\). This makes identifying other features of the ellipse straightforward, as its dimensions and orientation are clearly defined in relation to its center.Understanding this structure helps recognize that the ellipse stretches more in the y-direction, due to its larger denominator under the y-term, revealing its orientation and the nature of its axes.