Conic Sections in Polar Coordinates

Focus-Directrix Definitions of the Conic Sections Let F be a fixed point, the focus, and let D be fixed line, the directrix, in a plane. A conic section, or conic, is the set of all points P in the plane such that where e is a fixed positive number, called the eccentricity. If e = 1, the conic is a parabola. If e < 1, the conic is an ellipse. If e > 1, the conic is a hyperbola. PF PD = e

Focus-Directrix Definitions of the Conic Sections P F ˝/2 Directrix Focus at pole e = 1 Parabola O D P F ˝/2 Directrix Focus at pole e < 1 Ellipse O D P Directrix Focus at pole F ˝/2 e > 1 Hyperbola D´ P´

Standard Forms of the Polar Equations of Conics Let the pole be a focus of a conic section of eccentricity e with the directrix |p| units form the focus. The equation of these 4 conics are listed below. 1. Directrix: x = p Polar axis or x-axis  = /2 or y-axis Focus at pole Directrix: x = - p 2.

Standard Forms of the Polar Equations of Conics Let the pole be a focus of a conic section of eccentricity e with the directrix |p| units form the focus. The equation of these 4 conics are listed below. 3. Directrix: y = p Polar axis or x-axis  = /2 or y-axis Focus at pole Directrix: y = - p 4.

Graphing the Polar Equation of a Conic If necessary, write the equation in one of the standard forms. Use the standard form to determine values for e and p. Use the value of e to identify the conic. Use the appropriate figure for the standard form of the equation shown on the previous slide to help guide the graphing process.

Text Example Graph the polar equation: Solution Step 1 Write the equation in one of the standard forms. The equation is not in standard form because the constant term in the denominator is not 1. Solution To obtain 1 in this position, divide the numerator and denominator by 2. The equation in standard form is ep = 4 e = 1/2

Text Example cont. Solution Step 2 Use the standard form to find e and p, and identify the conic. The voice balloons show that e = 1/2 and ep = 1/2 p = 2. Thus, e = 1/2 and p = 4. Because e = 1/2 < 1, the conic is an ellipse. Solution Step 3 Use the figure for the equation’s standard form to guide the graphing process. The figure for the conic’s standard form is shown below. We have symmetry with respect to the polar axis. One focus is at the pole and a directrix is x =4, located four units to the right of the pole. Directrix: x = p Polar axis or x-axis  = /2 or y-axis Focus at pole

Text Example cont. Solution The last figure indicated that the major axis is on the polar axis. Thus, we find the vertices by selecting 0 and  for . The corresponding values for r are 4/3 and 4, respectively. The figure on the right shows the vertices (4/3, 0) and (4, ). Solution You can sketch the upper half of the ellipse by plotting some points from  = 0 to  = . Using symmetry with respect to the polar axis, you can sketch the lower half. 3 4 5  6 4 2 2 3 7 11 Directrix Focus Vertex (4, ) Vertex (4/3, 0) 3.5 3.1 2.7 2 r 5 /6 3/4 2 /3 /2 

Example Graph the polar equation: Solution: e=1 and ep = 3 so p=3 the shape is a parabola, the focus is at the pole and the directrix is y=3, the vertex is at =/2.

Example cont. Graph the polar equation:

Conic Sections in Polar Coordinates