Also, we have (1)
Pl is parallel to the x-axis, so Pl = DE (2)
Substitute (2) and (3) to equation (1), we have
Then PF = r
According to the definition of e, we have , so
When e <1, , so , that is, the coefficient of is greater than 0. so it is an equation of ellipse.
When e > 1, , equation (7) is
, so it is an equation of hyperbola.
Please see the attached file for the complete solution.
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4. The conversion between the polar and rectangular coordinates is
r + r e cos = ed, or r = ...
Conic Sections in Polar Coordinates are investigated.