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Conic Sections

Problems

Polar Form of the Conics

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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 = .

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