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Polar equations are a way to represent curves using a coordinate system based on angles and distances from a fixed point. Unlike the Cartesian coordinate system, which uses x and y axes, polar coordinates describe the position of a point as

• a distance ( ") from the origin (also known as the pole)
• an angle ( heta), measured from a reference direction

This system is particularly useful for dealing with conic sections that have their focus at the origin.
A conic section, such as an ellipse, parabola, or hyperbola, can be represented using a polar equation that depends on its eccentricity and directrix. The general polar form of a conic with its focus at the origin is written as:\[ r = \frac{ed}{1 + e \cos \theta} \] if the directrix is vertical, or as:\[ r = \frac{ed}{1 + e \sin \theta} \] if the directrix is horizontal.
This setup allows you to model conics like hyperbolas in a simple and intuitive way, making it easier to study their properties in polar coordinates.

A hyperbola is one of the distinct types of conic sections formed when a plane cuts through a double cone. It has two separate curves, which may look like mirror images of each other.When represented in a polar coordinate system, it assumes a unique form. The equation of a hyperbola in polar form involves the eccentricity \( e \), which is greater than 1 for hyperbolas, and the directrix distance \( d \).Key features of a hyperbola include:

• Two distinct branches, each asymptotically approaching a pair of lines.
• An eccentricity \( e > 1 \).
• A focal length that extends to infinity, as the branches continue indefinitely.

In the problem at hand, the hyperbola satisfies these conditions with an eccentricity of 4 and vertical directrix. Using the directrix \( x = 5 \), the polar equation becomes:\[ r = \frac{20}{1 + 4 \cos \theta} \]This equation fully describes the curve's properties in relation to the pole (origin) and provides an easy way to visualize the hyperbola's shape and positioning in polar coordinates.

Eccentricity is a crucial parameter in defining the shape of a conic section. It determines how "stretched" or "flattened" the conic appears.Here's how eccentricity values classify different conic shapes:

• If \( e = 0 \), the conic is a circle.
• If \( 0 < e < 1 \), the conic is an ellipse.
• If \( e = 1 \), the conic is a parabola.
• If \( e > 1 \), the conic is a hyperbola.

The exercise focuses on a hyperbola with an eccentricity of 4. This value means the hyperbola is significantly elongated compared to a circle or an ellipse.

Eccentricity measures how much the conic deviates from being circular. In a polar equation, eccentricity directly influences the curve’s equation form and its characteristics. For hyperbolas, larger eccentricity results in more spread-out branches. Understanding eccentricity helps not only in solving equations but also in visualizing the shapes and behaviors of the conic sections.