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10th Class Mathematics Quadratic Equations Question Bank Quadratic Equation

  • question_answer
    If \[\left( {{p}^{2}}-{{q}^{2}} \right){{u}^{2}}+\left( {{q}^{2}}-{{r}^{2}} \right)u+{{r}^{2}}-{{p}^{2}}=0\] and\[\left( {{p}^{2}}-{{q}^{2}} \right){{v}^{2}}+\left( {{r}^{2}}-{{p}^{2}} \right)v+{{q}^{2}}-{{r}^{2}}=0\] have a common root for p, q, r \[\in \] R and u, v being variables in the respective equations, find the common root.

    A)  ? 3                              

    B)  1        

    C)  3                                

    D)  ? 6

    Correct Answer: B

    Solution :

    (b): \[\left( {{p}^{2}}-{{q}^{2}} \right){{u}^{2}}+\left( {{q}^{2}}-{{r}^{2}} \right)u+\left( {{r}^{2}}-{{p}^{2}} \right)=0\] The sum of the coefficients \[\left( {{p}^{2}}-{{q}^{2}} \right){{u}^{2}}+\left( {{q}^{2}}-{{r}^{2}} \right)u+\left( {{r}^{2}}-{{p}^{2}} \right)=0\] \[\therefore u=1\]is a root of Eq. (1) \[\left( {{p}^{2}}-{{q}^{2}} \right){{v}^{2}}+\left( {{r}^{2}}-{{p}^{2}} \right)v+{{q}^{2}}-{{r}^{2}}=0\] \[{{p}^{2}}-{{q}^{2}}+{{r}^{2}}-{{p}^{2}}+{{q}^{2}}-{{r}^{2}}=0\]  \[\therefore v=1\]is a root of Eq. (2) \[\therefore 1\] is the common root of Eq. (1) and (2).   


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