Give the eccentricities of conic sections with one focus at the origin along with the directrix

step 1 In this question we have the eccentricity of a conic section E is equal to 5. Its one focus lie

Give the eccentricities of conic sections with one focus at...

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Key Concepts

Conic Sections

Conic sections are curves formed by the intersection of a plane with a double-napped cone. They can be classified as circles, ellipses, parabolas, or hyperbolas, and each type has distinct properties defined by the relationship between a fixed point (the focus) and a line (the directrix).

Eccentricity

Eccentricity is a non-negative number that measures the deviation of a conic section from being circular. It gives a quantitative description of the conic's shape: an eccentricity of 1 characterizes a parabola, less than 1 defines an ellipse, and greater than 1 produces a hyperbola.

Directrix

A directrix is a fixed line used in the geometric definition of a conic section. Along with a given focus, the directrix helps define the conic as the set of all points for which the ratio of the distance to the focus and the distance to the directrix remains constant (equal to the eccentricity).

Polar Equation of Conics

The polar equation of a conic section expresses the curve in a coordinate system where the focus is at the pole. In this form, the relationship between the radial distance and the angle is typically represented as r = ed / (1 + e cos?) (or a related variation depending on orientation), which directly reflects the fixed ratio defined by the eccentricity and the distances to the focus and directrix.

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