Problem :
Is the directrix of the conic horizontal or vertical? Does it lie to
the left, the right, above, or below the pole? Conic: r =
The directrix is horizontal, and below the pole.
Problem :
What type of conic is the following: r = ?
r = = .
e = > 1, so the conic is a hyperbola.
Problem :
What type of conic is the following: r = ?
r = = .
e = < 1, so the conic is an ellipse.
Problem :
Find the vertex (or vertices) of the conic r = .
e = 1, so the conic is a parabola, and it has a horizontal directrix above
the pole. Because its directrix is horizontal, its axis must be vertical.
So the vertex will occur on the line
θ = .
(r,) = (2,) is the vertex of the parabola. Note:
Another way to find the vertex is to use the fact that
p, the distance from
the focus to the directrix, is known to be
4 in this problem.
Problem :
Find a, b, and c of the conic r = .
e = 2, so the conic is a hyperbola. The directrix is vertical and to the left
of the pole. The transverse axis is horizontal. The vertices are at
(- 6, 0) and
(2, Π). So the transverse axis is
8 units long, so
a = 4.
Therefore
c = 8, and
b = 4.
Problem :
Express the parabola whose focus is the pole and whose horizontal directrix is 5
units above the pole in polar form.
r = .