Parallel to Polar Axis?
A ( 3 , 4 π ) and B ( 3 , 4 3 π ) are two points in a polar coordinate system.
And the line A B is parallel to polar axis.
True or False ?
Any line C D is parallel to the polar axis where C ( r 1 , θ 1 ) , D ( r 2 , θ 2 ) and r 1 = r 2 .
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2 solutions
If r 1 = r 2 and θ 1 ≡ θ 2 ( m o d 2 π ) then points A and B coincide so there is not really a line A B to begin with.
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Equally true if θ 1 + θ 2 = π in the case θ 1 = θ 2 = 2 1 π .
To be parallel to the polar axis, we must have the additional condition that θ 1 + θ 2 ≡ π ( m o d 2 π ) and θ i = 2 π ( m o d 2 π ) .
To be parallel to the polar axis, we need r 1 sin θ 1 = r 2 sin θ 2 . If r 1 = r 2 then we need sin θ 1 = sin θ 2 as well, so that either θ 1 ≡ θ 2 or else θ 1 + θ 2 ≡ π ( m o d 2 π ) .