Conic sections in polar coordinates

In summary, to find the directrix of a conic section in polar coordinates, you can use the eccentricity and the distances from the focus and vertex to the center of the ellipse. The distance from the center to the directrix is given by a2/c, where a is the distance from the center to the vertex and c is the distance from the center to the focus.
  • #1
glasshut137
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[SOLVED] conic sections in polar coordinates

Homework Statement



write a polar equation of a conic with the focus at the origin and the given data.

i know it's an ellipse with eccentricity 0.8 and vertex (1, pie/2)


The Attempt at a Solution



my question is: how do I find the directrix using the vertex and the focus point?
 
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  • #2
You can use the eccentricity, together with the distance from the focus to the vertex to determine where the center of the ellipse is. If the distance from the center to the focus is c and from the center to the vertex is a, then the distance from the center to the directrix is a2/c.
 

Related to Conic sections in polar coordinates

What are conic sections in polar coordinates?

Conic sections in polar coordinates are mathematical curves formed by intersecting a plane with a cone. They are described using polar coordinates, which use distance and angle measurements from a fixed point to represent points on a curve.

What are the 5 types of conic sections in polar coordinates?

The 5 types of conic sections in polar coordinates are: circles, ellipses, parabolas, hyperbolas, and degenerate conics.

How are conic sections in polar coordinates different from Cartesian coordinates?

In polar coordinates, points on a curve are described using distance and angle measurements from a fixed point, while in Cartesian coordinates, they are described using x and y coordinates. In polar coordinates, the focus and directrix of a conic section are represented by a fixed point and a line, respectively, while in Cartesian coordinates, they are represented by a pair of points or lines.

How are conic sections in polar coordinates used in real life?

Conic sections in polar coordinates have many real-life applications, including in astronomy to describe the orbits of planets and other celestial bodies, in engineering to design parabolic reflectors and satellite dishes, and in navigation to plot the distance and direction of objects from a fixed point.

What are some common equations for conic sections in polar coordinates?

The equations for conic sections in polar coordinates are:

  • Circle: r = a
  • Ellipse: r = a(1 - e cos θ) or r = a(1 + e cos θ)
  • Parabola: r = a(1 + cos θ) or r = a(1 - cos θ)
  • Hyperbola: r = a(1 + e cos θ) or r = a(1 - e cos θ)

Where a represents the distance from the origin to the focus, e represents the eccentricity, and θ represents the angle from the polar axis.

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