1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Converting A Polar Equation to Rectangular Form; Equation of a Circle

  1. Jul 22, 2010 #1
    1. The problem statement, all variables and given/known data
    Convert the polar equation

    r = 2(h cos θ + k sin θ)

    to rectangular form and verify that it is the equation of a circle. Find the radius and the rectangular coordinates of the center of the circle.

    2. Relevant equations

    3. The attempt at a solution

    First, I multiply both sides by r and distribute.


    apply the x= r cos θ and y= r sin θ equations

    from here I can factor out the 2 and plug it into the equation for a circle.


    not quite sure what do do from here.

    The answer to the problem is supposed to be:

    [tex](x-h)^2+(y-k)^2=h^2+k^2; \sqrt{h^2+k^2}; (h,k)[/tex]
  2. jcsd
  3. Jul 22, 2010 #2


    Staff: Mentor

    So far, so good. Separate the terms on the right, and bring them over to the left. Then complete the squares in the x and y terms.
  4. Jul 22, 2010 #3
    Hmm. Ok, I think I know what you mean.


    bring it over to the other side and complete the square and you get


    How would you get the [tex]h^2+k^2[/tex] on the RHS of the equation?
  5. Jul 22, 2010 #4


    Staff: Mentor

    You're skipping the steps that would produce what you're looking for.
    ==> x2 - 2hx +y2 - 2ky = 0

    Now, when you complete the squares in the x and y terms what do you need to add? You'll need to add the same amount on the right side.
  6. Jul 22, 2010 #5
    If it still isn't obvious what you need to add on each side of your equation, simply expand some squares to gain some insight.

    What about,

    [tex] (x+2)^{2} = x^{2} + 4x + 4 [/tex]


    [tex] (x + 8)^{2} = x^{2} + 16x + 64 [/tex]

    So now,

    [tex] x^{2} - 2hx + ? [/tex]

    What do you need to add (maybe in terms of h :wink:) in order to complete the square?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook