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Circle equation question

  1. May 14, 2008 #1
    1. The problem statement, all variables and given/known data
    A circle has equation x² + y² - 8x - 4y = 9

    (i) Show that the centre of this circle is (4,2) and find the radius of the circle.


    2. Relevant equations
    Circle equation = (x-a)² + (x-b)² = r²


    3. The attempt at a solution
    Well, if the centre of the circle is (4,2). Then the equation will be something like:

    (x-4)² + (y-2)² = 9

    except, if I expand out the two brackets, I get

    (x² -8x + 16) + (y² -4y + 4) = 9

    x² - 8x + y² -4y + 20 = 9

    Here is where i get confused

    So (i'm just double checking here since I don't have the answers), is the radius of the circle sqr9 or sqr 11? And is this really showing that (4,2) is the centre?

    Thanks
     
    Last edited: May 14, 2008
  2. jcsd
  3. May 14, 2008 #2

    Hootenanny

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    You have expanded the brackets incorrectly.
     
  4. May 14, 2008 #3
    oops, fix'd (i hope :) )
     
  5. May 14, 2008 #4

    Don't let the post above you put you off. You've expanded right but your fundamentally wrong.

    I'll explain this simply.

    x² + y² - 8x - 4y = 9

    ok you have to make the factorization to reach the -8x and the -4y

    which is (x-4)^2 + (y-2)^2 = 9 which is correct.

    Though this isn't finished, because when you factorized into those brackets you also added an extra -4^2 and a -2^2 that you didn't need. So to balance this, you have to add these to the right hand side of the equation.

    it ends up : (x-4)^2 + (y-2)^2 = 9 + 4^2 + 2^2
    = 29

    your radius is, for some reason, the square root of 29
     
  6. May 14, 2008 #5

    Hootenanny

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    No he didn't expand the brackets correctly, notice that he edited his post after I posted. See my quoted text above.
     
  7. May 14, 2008 #6
    sorry, I misread your post.

    I thought you told him he expanded correctly.
     
  8. May 14, 2008 #7
    Thanks for the replies, I managed to reach sqr29 using a different method though.

    x² + y² - 8x - 4y = 9

    so

    x² + y² - 8x - 4y - 9 = 0

    and the equation of a circle is

    (x - a)² + (x - b)² = r²

    expand out

    x² + y² - 2ax - 2by + a² + b² - r²

    equating coeficients

    -2a = -8
    a = 4

    -2b = -4
    b = 2

    constant terms

    a² + b² - r² = -9

    4² + 2² - r² = -9

    -r² = -29

    r = sqr29

    and I think that shows that the centre is (4,2) as well

    This way seems logical but much longer
     
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