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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.

    [tex]r^2=2hr\cos\theta+2kr\sin\theta[/tex]

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

    [tex]r^2=2hx+2ky[/tex]
    from here I can factor out the 2 and plug it into the equation for a circle.

    [tex]x^2+y^2=2(hx+ky)[/tex]

    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

    Mark44

    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.

    [tex]
    x^2+y^2=2hx+2ky
    [/tex]

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

    [tex](x-h)^2+(y-k)^2=0[/tex]

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

    Mark44

    Staff: Mentor

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

    or,

    [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?
     
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