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How to find the radii of these 2 circles given 2 known points

  1. Jun 19, 2017 #1
    1. The problem statement, all variables and given/known data


    Screenshot_34.png

    2. Relevant equations

    y-y1 = m (x-x1) ---> line equation

    $$ (x - a)^2 + (y-b)^2 = r^2 $$ ---> circle equation


    3. The attempt at a solution

    I tried to draw the triangles using, (1, 3) (2, 4) and (0, b)

    (0, b) is the tangent point to y-axis


    and used those points for making perpendicular bisector, so I can find the center of the circle, and find the distance from that center to one of the points, which that distance is the radii.


    between (1, 3) (2, 4)

    the mid point = 1.5 , 3.5 gradient = 1, the perpendicular m = -1

    the line equation = y - 1.5 = -1(x - 3.5) ..... (1)


    between (2,4) (0, b)

    $$ mid point = (1, \frac {4+b)} {2} )

    \\gradient = \frac {4-b}{2} \\the perpendicular gradient = \frac {-2}{4-b}

    \\the line equation = y - \frac {(4+b)} {2} = \frac {-2}{(4-b)} (x - 1) ...(2) $$


    between (1,3) (0,b)

    $$ \\mid point = (0.5, \frac {(3+b)} {2} )

    \\gradient = \frac {(3-b)}{1} \\the perpendicular gradient = \frac {-1}{(3-b)}

    \\the line equation = y - \frac {(3+b)} {2} = \frac {-1}{(3-b)} (x - 0.5) ...(3)

    $$


    if I find the x, y then its the center

    we know that the center (x, b) where b the same in b (0, b)

    but i don't think its work.
    can anyone give me another way to solve?

    I find another solutions but I don't get what the person doing can anyone explain it?
    https://www.algebra.com/algebra/homework/Circles/Circles.faq.question.1077722.html
     
  2. jcsd
  3. Jun 19, 2017 #2

    jbriggs444

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    Because we are asked not to graph this, I would rule out most approaches based on geometry and go for something algebraic.

    By inspection, the centers of both circles are at y=2. We do not know their respective x coordinates. So write down some equations.

    Let ##(x_1,y_1)## be the center of the first circle and let ##r_1## be its radius.

    The radius of the first circle is given by the distance from its center to the tangent point on the y axis:
    $$r_1 = x_1$$
    The radius of the first circle is also given by the distance from its center to point (1,3):
    $$r_1^2 = (1-x_1)^2 + (3-y_1)^2$$
    We already know that ##y_1 = 2##. We already have that ##r_1 = x_1##. Making those substitutions:

    $$x_1^2 = (1-x_1)^2 + 1^2$$

    Solve for ##x_1##

    Repeat for the second circle.
     
    Last edited: Jun 19, 2017
  4. Jun 19, 2017 #3
    how do you know the center y = 2? inspection of what?
     
  5. Jun 19, 2017 #4

    jbriggs444

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    My mistake. I completely misread the problem somehow.
     
  6. Jun 19, 2017 #5

    haruspex

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    You seemed to be doing ok. It should get you there.

    What do you not understand in the solution you linked?
     
  7. Jun 19, 2017 #6
    From the first, finding center of upper circle , how it comes up (1.5 -P/root 2 , 3.5 + P/root 2) ?
    And how to find radii? R^2 = (1.5 -P/root 2)^2 + (3.5 + P/root 2)^2 not the same answer as from the link

    While my method i get imaginary x
     
  8. Jun 19, 2017 #7

    haruspex

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    What is P?

    I would have done this:
    Centre of circle is at (R,b). This is distance R from each of the two given points....
     
  9. Jun 20, 2017 #8
    So i used your method
    $$ (R-1)^2 + (b-3)^2 = (R-2)^2 + (b-4)^2 $$
    R + b = 5
    b = 0 or 4
    R = 5 or 1
    So does it mean the center of circle A = R, b = 1, 4
    And circle B = R, b = 5, 0
    ??
     
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