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Math Help - Help with these problems please

  1. #1
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    Help with these problems please

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  2. #2
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    to #2:

    1. Use two different values of k: k = r and k = s

    2. You'll get two different circles:

    A: x^2 + y^2 + r\cdot x + (1 + r)\cdot y - (1 + r) = 0

    B: x^2 + y^2 + s\cdot x + (1 + s)\cdot y - (1 + s) = 0

    2. Calculate the coordinates of the common points (points of intersection) by subtracting columnwise:

    A-B: x(r - s) + y(r - s) - (r - s) = 0

    Divide by (r-s):

    x+y-1=0~\implies~x=1-y

    3. Plug in this term for x into the original equation:

    (1 - y)^2 + y^2 + k(1 - y) + (1 + k)y - (1 + k) = 0~\implies~ 2y^2 - y = 0

    Solve for y: y=0~\vee~y=\dfrac12

    4. Calculate the x-coordinate. You'll get 2 points whose coordinates are not depending on k, that means each circle passes through these two points: P_1(1,0)\ ,\ P_2\left(\dfrac12\ ,\ \dfrac12\right)
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