To find: The graph of the polar equation r = 13 − 2 cosθ(ellipse).
Solution:
Graph of the polar equation r = 13 − 2 cosθ(ellipse) is
Given:
It is asked to find the graph of the polar equation r = 13 − 2 cosθ(hyperbola):
Check for symmetry test:
Polar axis:
Replace θ by − θ. The result is r = 13 − 2 cos(− θ) = 13 − 2 cosθ.
The test satisfied. So the graph is symmetric with respect to the polar axis.
The line θ = π2:
Replace θ by π − θ. The result is
r = 13 − 2 cos(π − θ) = 13 − 2(cosπ cosθ + sinπ sinθ) = 13 − 2((− 1) cosθ + (0) sinθ)
= 13 + 2 cosθ
The test fails, so the graph may or may not symmetric with respect to the line θ = π2.
The Pole:
Replace r by − r. Then the result is − r = 13 − 2 cosθ, so r = − 13 − 2 cosθ. This test fails. Replace θ by θ + π. The result is
r = 13 − 2 cos(θ + π) = 13 − 2(cosπ cosθ − sinπ sinθ) = 13 − 2(− 1)cosθ = 13 + 2 cosθ.
This test also fails, so the graph may or may not be symmetric with respect to the pole.
Let’s identify points on the graph by assigning values to the angle θ and calculating the corresponding values of r.
As the polar equation is symmetric about the polar axis, it is enough to find the values from 0 to π.
To sketch:
r = 13 − 2 cosθ
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