Find a polar equation for each conic. For each, a focus is at the pole. e=6 ; directrix is paral

step 1 Given focus is at the pole is equal to six and the position of the directrix. We have to find a

Find a polar equation for each conic. For each, a focus is...

Key Concepts

Directrix

The directrix is a fixed line used in the focus-directrix definition of a conic section. Its position relative to the focus determines the orientation of the conic and influences the structure of its equation. When expressed in polar coordinates, the directrix’s orientation (for example, being horizontal or vertical) dictates whether the trigonometric function in the equation is sine or cosine.

Polar Coordinates

Polar coordinates represent points in the plane in terms of a distance from a fixed origin (radius, r) and an angle (?) relative to a fixed direction. Placing the focus of a conic at the pole often simplifies its equation, making it easier to derive and analyze using the polar form.

Standard Polar Conic Equation

The polar equation of a conic section with the focus at the pole is typically written as r = ed / (1 + e cos?) or r = ed / (1 + e sin?), depending on the orientation of the directrix. Here, d is the perpendicular distance from the focus to the directrix, and the choice between cosine and sine reflects whether the directrix is vertical or horizontal relative to the polar axis.

Focus-Directrix Property

This property describes conic sections as the set of all points for which the ratio of the distance to a fixed point (called the focus) to the distance to a fixed line (called the directrix) is constant. This constant ratio is known as the eccentricity, and the focus-directrix definition is fundamental in deriving the equations for conics in different coordinate systems.

Eccentricity

Eccentricity (e) is a parameter that determines the type and shape of a conic section. When e = 1, the conic is a parabola; when e < 1, it is an ellipse; and when e > 1, it is a hyperbola. It directly measures the deviation of the conic from being circular, playing a central role in the conic’s defining equation.

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