For each equation: a) Tell whether the equation describes a...

For each equation: a) Tell whether the equation describes a parabola, an ellipse, or a hyperbola. b) State whether the directrix is vertical or horizontal and give its location in relation to the pole. c) Find the vertex or vertices. d) Graph the equation. $$r=\frac{12}{3+9 \cos \theta}$$

For each equation:
a) Tell whether the equation describes a parabola, an ellipse, or a hyperbola.
b) State whether the directrix is vertical or horizontal and give its location in relation to the pole.
c) Find the vertex or vertices.
d) Graph the equation.
$$r=\frac{12}{3+9 \cos \theta}$$

Key Concepts

Graphing Conics

Graphing a conic section given in polar coordinates involves plotting points using the polar equation and identifying key components such as the focus (pole), vertex (or vertices), and directrix. Understanding the conic’s eccentricity and symmetry is essential to accurately sketching its shape, and often converting to Cartesian coordinates or determining specific r and theta values can aid in producing a precise graph.

Vertex

The vertex of a conic section is the point that lies closest to or farthest from the focus along the axis of symmetry. In polar coordinates, the vertex is found by identifying the angle that minimizes or maximizes the value of r, which is especially straightforward when the polar conic equation is aligned with the reference axes.

Conic Eccentricity

Eccentricity (e) is a parameter that determines the shape of a conic section. For values of e less than 1, the conic is an ellipse; for e equal to 1, the conic is a parabola; and for e greater than 1, it is a hyperbola. In polar equations, the value of e directly influences the distance from the focus (the pole) to any point on the curve relative to its distance from the directrix.

Polar Conic Sections

Polar conic sections are represented by equations in polar coordinates, typically written in the form r = ed/(1 + e*cos(theta)) or r = ed/(1 + e*sin(theta)). This general form allows the representation of ellipses, parabolas, and hyperbolas by adjusting the eccentricity (e) and directrix distance (d), and it highlights the role of the pole as one of the key points in the conic’s geometry.

Directrix

The directrix is a fixed line used in the definition of a conic section. It works together with the focus (the pole in polar coordinates) such that the ratio of the distance of any point on the conic to the focus and its distance to the directrix is constant (equal to the eccentricity). Its orientation (vertical or horizontal) depends on the structure of the polar conic equation, typically based on whether the cosine term or sine term is present.

Transcript

Please give Ace some feedback