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Thread: Simultaneous equation

  1. #1
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    Simultaneous equation

    $\displaystyle \frac{x+\sqrt{a^2-x^2}}{x-\sqrt{a^2-x^2}}=b$

    Solve to find x.
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  2. #2
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    Hello, BabyMilo!

    Solve for $\displaystyle x\!:\;\;\frac{x+\sqrt{a^2-x^2}}{x-\sqrt{a^2-x^2}}\:=\:b$

    We have: .$\displaystyle x + \sqrt{a^2-x^2} \;=\;bx - b\sqrt{a^2-x^2} \quad\Rightarrow\quad b\sqrt{a^2-x^2} + \sqrt{a^2-x^2} \;=\;bx - x$


    Factor: .$\displaystyle \sqrt{a^2-x^2}(b+1) \;=\;x(b-1)$


    Square: .$\displaystyle (a^2-x^2)(b^2+2b + 1) \;=\;x^2(b^2-2b + 1)$


    Expand: .$\displaystyle a^2b^2 + 2a^2b + a^2 - b^2x^2 - {\color{red}\rlap{////}}2bx^2 - x^2 \;=\;b^2x^2 - {\color{red}\rlap{////}}2bx^2 + x^2$


    We have: .$\displaystyle 2b^2x^2 + 2x^2 \;=\;a^2b^2 + 2a^2b + a^2$


    Factor: .$\displaystyle 2x^2(b^2+1) \;=\;a^2(b^2+2b+1)$


    Hence: .$\displaystyle x^2 \;=\;\frac{a^2(b+1)^2)}{2(b^2+1)}$


    Therefore: .$\displaystyle x \;=\;\pm\frac{a(b+1)}{\sqrt{2(b^2+1)}} $

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