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Polar equations are a way to represent curves using a coordinate system based on radius (\(r\)) and angle (\(\theta\)). This is different from the Cartesian system where you use \(x\) and \(y\) coordinates. In polar coordinates:

• The origin is called the pole.
• The horizontal axis from the pole is the polar axis.

When you have an equation like \(r = \frac{3}{2 - 2 \sin \theta}\), it tells us how the radius \(r\) changes based on the angle \(\theta\). These equations are especially useful because they provide a concise way to describe conic sections like ellipses, parabolae, and hyperbolae.
By comparing this equation with the general form \(r = \frac{ed}{1 - e \sin \theta}\), we can identify the type of conic section. The important parameter here is the eccentricity (\(e\)), which determines the conic's shape.

Conic sections are curves obtained by intersecting a plane with a cone. The shape of the conic depends on the angle and position of the intersection. There are several types of conic sections:

• Circle: Occurs when the plane cuts the cone parallel to the base.
• Ellipse: Formed when the plane cuts through the cone at an angle, but not steep enough to be parallel.
• Parabola: Result from the plane being parallel to a cone's side.
• Hyperbola: Created when the plane cuts through both nappes of the cone.

Eccentricity (\(e\)) is key to classifying conic sections:

• Eccentricity = 0: Circle
• 0 < Eccentricity < 1: Ellipse
• Eccentricity = 1: Parabola
• Eccentricity > 1: Hyperbola

In our exercise, the eccentricity is 1, thus confirming that the conic section is a parabola.

A parabola is a special conic section characterized by being symmetric and having an eccentricity (\(e\)) of exactly 1. It is defined as the set of all points that are equidistant from a fixed point called the "focus" and a line called the "directrix".
In polar coordinates, the parabola will typically have the equation form \(r = \frac{ed}{1 - e \sin \theta}\). This becomes a parabola when the factor \(e\) equals 1. Let's consider the equation \(r = \frac{3}{2 - 2 \sin \theta}\).This matches the general parabola form since:

• The coefficient of \(\sin \theta\) is 2, equating to our \(e = 1\).
• The form is confirmed by the comparison with \(1 - e \sin \theta\).

Thus, the conic is confirmed to be a parabola. To visualize this, sketching the graph involves identifying key points: the vertex is at \((\frac{3}{2}, 0)\), demonstrating the parabola's symmetry around the vertical axis. Understanding the geometric properties helps in comprehending how this curve behaves and is represented in various applications.