Match the polar equation with the correct graph. [The graphs are labeled (a), (b), (c), (d), (e)

step 1 In this question, we are given the polar equation r is equal to 2 over 1 plus sign of theta, 1 p

Match the polar equation with the correct graph. [The graphs...

Key Concepts

Polar Coordinates

Polar coordinates provide a method of representing points in the plane by using a distance from the origin (the radius) and an angle from a reference direction. This system is particularly useful for equations that have inherent rotational symmetry or circular features, and it simplifies the representation and analysis of curves like circles and conic sections that may be more complicated in Cartesian coordinates.

Conic Sections in Polar Form

Many conic sections such as ellipses, parabolas, and hyperbolas can be expressed in polar form where the focus is at the origin. The general polar equation for a conic is of the form r = ed/(1 + e sin?) or r = ed/(1 + e cos?), where e is the eccentricity and d is a constant related to the directrix. This formulation directly relates the geometric properties of the conic (like its shape and orientation) to the equation.

Eccentricity

Eccentricity (denoted by e) is a parameter that defines the shape of a conic section. For a parabola, the eccentricity is exactly 1; for an ellipse it is less than 1; and for a hyperbola, it is greater than 1. Recognizing the eccentricity in a polar equation helps in identifying the type of conic section represented by the equation.

Graphical Analysis of Polar Equations

Graphing polar equations involves understanding how the radius r changes with the angle ?. For conic sections, rewriting the equation in polar form reveals features such as the focus at the pole and the orientation of the directrix. This aids in visual matching of the equation to its graph by identifying features like the opening direction and the general shape of the curve.

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