Line passing through the origin (polar coordinates)

[Poetria]

Homework Statement

-infinity < r > +infinity

Which of the following are equations for the line y=m*x for m<0:

a. theta = -arctan(m)
b. theta = arctan(m)
c. theta = arctan(-m)
d. theta = arctan(m) + pi
e. theta = arctan(m) - pi
f. r = 1/(sin(theta - arctan(m)))

2. The attempt at a solution

I think c. theta = arctan(-m) is the solution.

As m tends to negative infinity theta approaches the limit -pi/2. But for a. and b. theta approaches pi/2.

d. and e. theta differs from the desired limit.

f. does not apply to the line passing through the origin.
y=m*x+b, for b not equal to 0

Is my reasoning correct?

 

[scottdave]

Pick a slope (say m = 1). What is the associated angle for this slope? Which one gives this angle?

Some of them could be point to the same line, since r can be a negative number.

Is this something where you have the option to pick more than one of the choices?

 

[Poetria]

Pick a slope (say m = 1). What is the associated angle for this slope? Which one gives this angle?

Some of them could be point to the same line, since r can be a negative number.

Is this something where you have the option to pick more than one of the choices?

Yes, you can pick more options in this exercise.

I haven't got what you mean about r because it disappears from the final equation. Hm I have to think about it. :(

Would also theta=-arctan(m) do? Hm, I could pick 1 but this is a positive number.
-pi/4=-arctan(1)

 

[scottdave]

When r is not present in the expression, then you must have a graph which accommodates all values of r. It is like in rectangular coordinate you have y=0, this is a line where x can be any value

 

[scottdave]

A slope of 1 and -1are good ones to start with, but shouldn't be the only tries, but try some others like 3/4 or 4/3. You don't need to actually calculate an angle to see if it will work. Try drawing a picture

 

[Poetria]

I think I have a problem with the concept of a negative radius. I wasn't familiar with it at all.

So perhaps the options:
d. theta = arctan(m) + pi
e. theta = arctan(m) - pi

are correct provided the radius is negative.

I have found a picture here:
http://tutorial.math.lamar.edu/Classes/CalcII/PolarCoordinates.aspx

 

[scottdave]

Rather than a negative "radius", think about moving in the negative direction. For example, point your car South, put in reverse and you move North. It is similar thinking here. Polar coordinate always specify a point by a distance away from the origin, but if you think about getting to each point with the car example, it should help.

 

[scottdave]

So for the straight line, point your car at a certain angle, then move forward or backwards toward infinity. There is more than one angle which puts your car on that same line

 

[Poetria]

Great explanation. :) Many thanks. :)

 

[scottdave]

I think I have a problem with the concept of a negative radius. I wasn't familiar with it at all...

http://tutorial.math.lamar.edu/Classes/CalcII/PolarCoordinates.aspx

I like that site.

 

[scottdave]

So what do you think are the correct answer choices, now?

 

[Poetria]

The correct ones:

b. theta = arctan(m)
For negative radii:
d. theta = arctan(m) + pi
e. theta = arctan(m) - pi

Yeah, this site is very helpful. :)