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Math Help - Determine the centre and radius of a circle

  1. #1
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    Determine the centre and radius of a circle

    Given is the following equation of a circle, for which I've to determine its centre and radius. I'm running into some difficulties when I'm trying to complete the squares.

    x^2 + y^2 -2x + 4y = 4 \Leftrightarrow (x - 1)^2 + (y + 2)^2 = 4
    This leads me to believe the centre would be (1, -2). Now to find its radius I'll just complete the squares and simplify:
    (x - 1)^2 + (y + 2)^2 = 4 \Leftrightarrow x^2 - 2x + 1 + y^2 + 4y + 4 = 4 \Leftrightarrow  x^2 - 2x + y^2 + 4y = -1

    This obviously isn't correct, -1 being kind of a weird radius . So what am I doing wrong here? The correct answer (according to my textbook) should be r=3. I guess I'm missing something with completing the y-part, since -1 + 4 = 3, but can't wrap my head around it...

    Any handy tips for solving these kind of exercises?
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  2. #2
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    Re: Determine the centre and radius of a circle

    Quote Originally Posted by Lepzed View Post
    Given is the following equation of a circle, for which I've to determine its centre and radius. I'm running into some difficulties when I'm trying to complete the squares.

    x^2 + y^2 -2x + 4y = 4 \Leftrightarrow (x - 1)^2 + (y + 2)^2 = 4
    This leads me to believe the centre would be (1, -2). Now to find its radius I'll just complete the squares and simplify:
    (x - 1)^2 + (y + 2)^2 = 4 \Leftrightarrow x^2 - 2x + 1 + y^2 + 4y + 4 = 4 \Leftrightarrow  x^2 - 2x + y^2 + 4y = -1

    This obviously isn't correct, -1 being kind of a weird radius . So what am I doing wrong here? The correct answer (according to my textbook) should be r=3. I guess I'm missing something with completing the y-part, since -1 + 4 = 3, but can't wrap my head around it...

    Any handy tips for solving these kind of exercises?
    Your problem is that you're saying that (x-1)^2 = x^2-2x when it does not. The answer is x^2-2x = x^2 - 2x +1 - 1 = (x-1)^2 - 1

    In other words once you've completed the square on the LHS you should have (x-1)^2 -1 + (y+2)^2 - 4 = 4 and you can then add 5 to both sides to remove the constants: (x-1)^2 + (y+2)^2 = 9 which gives the radius as 3.

    Your co-ordinates for the centre of the circle are correct
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    Re: Determine the centre and radius of a circle

    Ah, I see! Thanks!
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  4. #4
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    Re: Determine the centre and radius of a circle

    I'm practicing some more and stumbled upon the following: describe the region defined by the following pair of inequalities:

    x^2 + y^2 < 2x,  x^2 + y^2 < 2y
    I'm having some difficulties interpreting this. At first I thought I could rewrite each inequality to try to make some more sense of it, like so: x^2 + y^2 < 2x \Leftrightarrow x^2 + y^2 - 2x < 0, but that still doesn't make alot of sense to me.

    What I've further tried, is to just write it as a 'normal' equation: x^2 + y^2 = 2x, then x & y are \sqrt2, same for X^2 + y^2 = 2y. Don't these circles describe exactly the same area?
    If not, how should I deal with the 2x and 2y?
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  5. #5
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    Re: Determine the centre and radius of a circle

    Quote Originally Posted by Lepzed View Post
    x^2 + y^2 < 2x,  x^2 + y^2 < 2y
    I'm having some difficulties interpreting this. At first I thought I could rewrite each inequality to try to make some more sense of it, like so: x^2 + y^2 < 2x \Leftrightarrow x^2 + y^2 - 2x < 0, but that still doesn't make alot of sense to me.
    That system can be rewritten by completing squares.
    \left\{ \begin{gathered}  (x - 1)^2  + y^2  < 1 \hfill \\  x^2  + (y - 1)^2  < 1 \hfill \\ \end{gathered}  \right.

    Those are both the interiors of circles, open disks.
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