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Math Help - Showing three points create a circle (forming a 3x3 linear system?)

  1. #1
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    Showing three points create a circle (forming a 3x3 linear system?)

    Well, yeah been kinda active here recently so most of you know the ordeal me and my classmates are in. And being that our teacher is of ZERO help, you guys have been AMAZING for us. Once again sorry for asking so many questions, but we REALLY need the help.

    anyway, we have a bunch of these, heres is an example of one:

    1)Show the points (2,6), (-5,-1), and (-2,8) determine a circle. A graph is NOT proof (Hint:general form, 3x3 linear system). Also, find the standard form of this circle.

    We then have alot of point variations, but i assume the process for them is all the same.

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by Danktoker View Post
    Well, yeah been kinda active here recently so most of you know the ordeal me and my classmates are in. And being that our teacher is of ZERO help, you guys have been AMAZING for us. Once again sorry for asking so many questions, but we REALLY need the help.

    anyway, we have a bunch of these, heres is an example of one:

    1)Show the points (2,6), (-5,-1), and (-2,8) determine a circle. A graph is NOT proof (Hint:general form, 3x3 linear system). Also, find the standard form of this circle.

    We then have alot of point variations, but i assume the process for them is all the same.

    Thanks in advance.
    1. The equation of a circle around M(x_M, y_M) with radius r is:

    (x-x_M)^2+(y-y_M)^2=r^2

    2. Plug in the coordinates of the given points. You'll get a system of equations:

    \left|\begin{array}{rcl}(2-x_m)^2+(6-y_M)^2&=&r^2 ~~~\bold{\color{blue}\text{[1]}}\\<br />
(-5-x_m)^2+(-1-y_M)^2&=&r^2 ~~~\bold{\color{blue}\text{[2]}}\\<br />
(2-x_m)^2+(8-y_M)^2&=&r^2~~~\bold{\color{blue}\text{[3]}} \end{array}\right.

    3. Expand the brackets. Afterwards calculate [1] - [2] and [2] - [3]. You'll get a system of simultaneous equations in x_M and y_M.

    4. Plug in the coordinates of the center into any of the 3 equations to calculate the length of r.

    Spoiler:
    I've got M(-2, 3) and r = 5
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  3. #3
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    Thank you so much.

    I was finally able to figure all of this out.

    You have made my weekend.
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