Identify each of the following as the equations of an...

Identify each of the following as the equations of an ellipse, a hyperbola, or a parabola: (a) $r=\frac{12}{3-\cos \theta}$; (b) $r=\frac{12}{1+3 \cos \theta}$; (c) $r=\frac{12}{1+\cos \theta}$; (d) $r=\frac{12}{3-8 \cos \theta}$ Ans. (a) ellipse; (b) hyperbola; (c) parabola; (d) hyperbola

Identify each of the following as the equations of an ellipse, a hyperbola, or a parabola:
(a) $r=\frac{12}{3-\cos \theta}$;
(b) $r=\frac{12}{1+3 \cos \theta}$;
(c) $r=\frac{12}{1+\cos \theta}$;
(d) $r=\frac{12}{3-8 \cos \theta}$
Ans. (a) ellipse; (b) hyperbola; (c) parabola; (d) hyperbola

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Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. They include ellipses, parabolas, and hyperbolas, each with its own distinctive geometric properties. In the context of polar coordinates, these curves are often described in a unified form that highlights a key parameter—the eccentricity—which directly determines the type of conic.

Eccentricity

Eccentricity is a crucial parameter that defines the shape of a conic section. It is the ratio that compares the distance of any point on the conic from its focus to its distance from the directrix. When the eccentricity (denoted by e) is less than 1, the conic is an ellipse; when it is equal to 1, the conic is a parabola; and when it is greater than 1, the conic is a hyperbola.

Polar Equations of Conics

In polar coordinates, conic sections with a focus at the pole (origin) are typically written in the form r = ed/(1 + e cos?) or r = ed/(1 + e sin?), where d is the distance to the directrix and e is the eccentricity. This representation simplifies the classification of conic sections, as the type of conic can be determined directly by the value of the eccentricity.

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