Eccentricity
Directrix
The directrix is a fixed line. In the case of the ellipse, the directrix is parallel to the minor axis and perpendicular to the major axis. In the picture to the right, the distance from the center of the ellipse (denoted as O or Focus F; the entire vertical pole is known as Pole O) to directrix D is p.
Directrices may be used to find the eccentricity of an ellipse.
Directrices may be used to find the eccentricity of an ellipse.
Formula for finding r of an ellipse in polar form
As you may have seen in the diagram under the "Directrix" section, r is not the radius (as ellipses don't have radii). Rather, r is the value from any point P on the ellipse to the center O.
The formula for finding the value r is:
r= ep/(1+ecosθ)
Proof:
The formula for finding the value r is:
r= ep/(1+ecosθ)
Proof:
- Start with the formula for eccentricity.
- Note that according to the diagram under the "Directrix" section, the distance from point D to point P is the same as the distance p + the distance from O to Q. The distance from O to Q can be rewritten as rcos(theta) because cos(theta) = d(O, Q)/r so rcos(theta) = d(O, Q).
- Go back to the formula for eccentricity and multiply d(D, P) on both sides.
- Substitute d(D, P) for p + rcos (theta) because of the reasoning in step 2.
- Distribute.
- Subtract ercos(theta) on both sides.
- On the side with r - ercos(theta), factor out r.
- Divide 1 - ecos(theta) on both sides.
To sketch an ellipse, simply substitute special value points (0, pi/2, pi, 3pi/2) into the equation for finding r.
Converting equations of ellipses from rectangular to polar form
Firstly, remember the rules for converting from rectangular to polar:
To convert a rectangular equation into polar form, remove the numerators. Then, start changing rectangular values into polar form as per the rules above. Keep solving until you isolate the variable r.
- x = rcos (theta)
- y = rsin (theta)
- r = sq. rt. (x^2 + y^2)
- theta = tan^-1 (y/x)
To convert a rectangular equation into polar form, remove the numerators. Then, start changing rectangular values into polar form as per the rules above. Keep solving until you isolate the variable r.