Convert x^2+y^2=4y-2x to Polar Equation

In summary, the equation in polar coordinates for the curve is r^2 = 4rsin@ - 2rcos@, and it can be simplified by dividing both sides by r. However, it is important to note that this should not be done if r happens to equal zero.
  • #1
Calcgeek123
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Homework Statement


Establish an equation in polar coordinates for the curve x^2+y^2=4y-2x


Homework Equations



n/a

The Attempt at a Solution


I know that x^2+y^2=r^2 so I used substitution, and now have r^2=4y-2x. Now this next part, I'm really not sure if I'm allowed to do this... i know that x=rcos@ (let @=theta) and y=rsin@. So i simply substituted to get r^2=4rsin@-2cos@. Is that allowed? Also, i feel like my answer should just look like r=... So should I take the sqaure root of both sides?
Thank you, very appreciated.
 
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  • #2
You should have r^2 = 4rsin@ - 2rcos@. You lost an "r" on your cosine term. You can simplify by dividing both sides by r, but you don't want to take the square root of both sides. You normally have to be careful of dividing by r, in case r happens to be equal to zero. That's not a problem in this case, since the curve doesn't go through the origin.
 
  • #3
Yeah, the missing r is a type. Dividing makes sense though, duh. thanks so much!
 

FAQ: Convert x^2+y^2=4y-2x to Polar Equation

1. What exactly is a polar equation?

A polar equation is a way of representing a curve or shape on a polar coordinate system, where points are defined by a distance from the origin and an angle from a fixed reference line.

2. How do I convert a rectangular equation to a polar equation?

To convert a rectangular equation, such as x^2+y^2=4y-2x, to a polar equation, you can use the following steps:

  1. Replace x with rcosθ and y with rsinθ. This will give you an equation in terms of r and θ.
  2. Simplify the equation using trigonometric identities.
  3. Use the Pythagorean identity (r^2 = x^2 + y^2) to eliminate r^2 from the equation.
  4. If needed, rearrange the equation to solve for r in terms of θ.

3. Can all rectangular equations be converted to polar equations?

No, not all rectangular equations can be converted to polar equations. Some equations may not have a polar equivalent or may result in a complex or undefined equation.

4. What is the advantage of using a polar equation over a rectangular equation?

Polar equations can sometimes be simpler and more elegant than their rectangular counterparts, making them easier to work with and visualize on a polar coordinate system. They are also useful for representing shapes with circular or radial symmetry.

5. How can I graph a polar equation?

To graph a polar equation, you can plot points by substituting different values for θ and r and then connecting them. You can also use a graphing calculator or online graphing tool to visualize the equation. Keep in mind that the scale on a polar graph is different from a rectangular graph, with the radius increasing by a constant amount for each angle increment.

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