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Math Help - HELP with some circles

  1. #1
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    HELP with some circles

    Find the center and the radius of the circle x^2 + y^2 - 2ax + 4by = 0.

    ----

    Does every equation of the form x^2 + mx + y^2 + ny = p represent a circle? Explain.
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  2. #2
    Super Member Matt Westwood's Avatar
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    Complete the square on x and y and gather terms into something like:

    (x+p)^2 + (y+q)^2 = r^2

    Then (-p, -q) is the centre and r is the radius.

    It's not a circle if r^2 = 0 in which case it's a single point.

    Or when r^2 is negative in which case there are no points at all.
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  3. #3
    Senior Member bkarpuz's Avatar
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    Exclamation

    Quote Originally Posted by pyrosilver View Post
    Find the center and the radius of the circle x^2 + y^2 - 2ax + 4by = 0.

    ----

    Does every equation of the form x^2 + mx + y^2 + ny = p represent a circle? Explain.
    Let A,B,C\in\mathbb{R},
    then
    x^{2}+y^{2}+Ax+By+C=0
    represents a circle provided that
    A^{2}+B^{2}-4C>0
    and the circle is centered at
    \bigg(-\frac{A}{2},-\frac{B}{2}\bigg)
    and the radius is
    \frac{\sqrt{A^{2}+B^{2}-4C}}{2}.

    Also you may try squaring the given equation to get (x-m)^{2}+(y-n)^{2}-r^{2}=0, which is centered at (m,n) with a radius of r.
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