91Ó°ÊÓ

Understanding Cartesian Coordinates is crucial for transforming equations from a polar to a Cartesian form, as seen in the given exercise. These coordinates are defined on a two-dimensional plane using two perpendicular axes: the horizontal x-axis and the vertical y-axis.

In their simplest form, any point on this plane can be described as a pair \(x, y\). Here, the x-coordinate indicates how far left or right a point is, and the y-coordinate tells us how far up or down it is.

For converting polar coordinates (like the equation \(r = \frac{3}{2+4 \sin \theta}\)) into Cartesian form, we use the relationships:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

By leveraging these formulas, the equation was rewritten in terms of x and y. This allows us to analyze and graph it on a standard Cartesian plane.

Ellipses are a type of conic section that can be identified by their characteristic shape—elongated circles. From the transformed equation \(x^2 + 4y^2 = 9\), we identify it as an ellipse. This is because it fits the general form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

In our equation:

• \(a^2 = 9\), hence, \(a = 3\)
• \(b^2 = \frac{9}{4}\), meaning \(b = \frac{3}{2}\)

These values indicate the semi-major axis (3) aligns with the x-axis and the semi-minor axis (\(\frac{3}{2}\)) with the y-axis.

This centered ellipse at the origin provides a clear and symmetrical graph and understanding its properties helps in predicting behaviors of systems that can be modeled elliptically.

Polar Coordinates provide an alternative way to describe locations on a plane using a distance and an angle, rather than x and y values. In the exercise, the original equation \(r = \frac{3}{2+4 \sin \theta}\) uses polar coordinates.

Here:

• \(r\) is the radius or distance from the origin
• \(\theta\) is the angle from the positive x-axis

This system is especially useful for curves and shapes like circles and spirals, which naturally align to circular or rotational symmetries.

When converting polar to Cartesian, we translate \(r\) and \(\theta\) into \(x\) and \(y\) using the relationships:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)

This conversion was critical in our original problem, moving us from a polar equation to one recognizable as an ellipse on a Cartesian plane.