Polar equation of conics

In summary, the given polar equation r = 15/[3-2cos(theta)] can be rearranged to 5/[1-(2/3)cos(theta)], which represents an ellipse with e=2/3. The vertices of this ellipse can be found by plugging in certain values of θ and converting them to (x,y) coordinates. In this case, the vertices are located at (15,0) and (3,π). If the ellipse was not symmetric about the x or y axis, finding the axis of symmetry would be the first step in determining the vertices.
  • #1
Homework Statement
Conic: r = 15/[3-2cos(theta)]
Relevant Equations
conic general form when vertical directrix to left of pole:
r = ed/[1-e*cos(theta)], where d is the distance between focus at pole and the directrix
The text says that the following conic, r = 15/[3-2cos(theta)], can be rearranged to 5/[1-(2/3)cos(theta)]. The graph of the conic is an ellipse with e=2/3.
Then it says that the vertices lie at (15,0) and (3,pi). How did they find the vertices? Thanks.
 
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  • #2
hello I_love_science. Here is a graph of your polar equation for reference.

Screen Shot 2021-01-20 at 11.56.03 PM.png


If you notice the ellipse is symmetric about an axis, you can plug in certain values of ##θ## to determine the vertices in polar coordinates ##(r, θ)## and convert them to ##(x, y)## coordinates. In this example, the ellipse is symmetric about the x axis, which makes finding vertices more streamlined.

If the ellipse was not symmetric about the x or y axis, then my first step would be to find the axis of symmetry.
 

1. What is a polar equation of a conic?

A polar equation of a conic is a mathematical representation of a conic section (such as a circle, ellipse, parabola, or hyperbola) in polar coordinates. It relates the distance and angle of a point on the conic to its focus and directrix.

2. How is a polar equation of a conic different from a Cartesian equation?

A polar equation of a conic uses polar coordinates (r and θ) to describe the location of points on the conic, while a Cartesian equation uses rectangular coordinates (x and y). This means that the polar equation takes into account both distance and angle, while the Cartesian equation only considers distance.

3. What is the general form of a polar equation of a conic?

The general form of a polar equation of a conic is r = e/(1 ± ecosθ), where e is the eccentricity of the conic. This form is used for all conic sections except for the circle, which has a simpler form of r = a, where a is the radius.

4. How do you determine the type of conic represented by a polar equation?

The type of conic represented by a polar equation can be determined by the value of the eccentricity (e). If e = 0, the equation represents a circle. If 0 < e < 1, it represents an ellipse. If e = 1, it represents a parabola. And if e > 1, it represents a hyperbola.

5. What are some applications of polar equations of conics?

Polar equations of conics have various applications in physics, astronomy, and engineering. For example, they can be used to describe the orbits of planets and satellites, the shape of lenses and mirrors, and the path of particles in electric and magnetic fields.

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