11th Class Mathematics Sample Paper Mathematics Olympiad - Sample Paper-3

  • question_answer
    The equation of the common tangent to the curves \[{{y}^{2}}=8x\] and \[\mathbf{xy}\text{=}-\mathbf{1}\]is-

    A)  \[2y=x+8\]                  

    B)  \[y=x+2\]     

    C)  \[y=2x+1\]      

    D)  \[3y=9x+2\]

    Correct Answer: B

    Solution :

    [b] Equation of the tangent of the curve \[{{y}^{2}}=8x\,be\,y=mx+\frac{2}{m}\] And it must satisfy the equation \[-xy=-1\] \[\Rightarrow x\left( mx+\frac{2}{m} \right)=-1\Rightarrow m{{x}^{2}}+\frac{2}{m}x+1=0\] Which is quadratic equation in x and its roots be equal. i.e. D=0 \[\Rightarrow {{b}^{2}}-4ac=0{{\left( \frac{2}{m} \right)}^{2}}-4.m.1=0\] \[\Rightarrow {{m}^{3}}=1\] \[\therefore m=1\] Hence, equation of the comment tangent be \[y=x+2\] Hence option [b] is correct.

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