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

  • question_answer
    The value of k for which the circles \[{{x}^{2}}+{{y}^{2}}-~3x+ky-5=0\]and \[4{{x}^{2}}+4{{y}^{2}}-12x-y-9=0\]became concentric is

    A)  \[-\frac{1}{6}\]                        

    B)  \[\frac{1}{6}\]             

    C)  \[-\frac{1}{4}\]                        

    D)  \[\frac{1}{4}\]

    Correct Answer: C

    Solution :

    [c]  \[{{S}_{1}}\equiv {{x}^{2}}+{{y}^{2}}-3x+ky-5=0.\] \[{{S}_{1}}\equiv {{x}^{2}}+{{y}^{2}}-3x-\frac{y}{4}-\frac{9}{4}=0.\] \[{{S}_{1}}\text{ }\!\!\And\!\!\text{ }{{S}_{2}}\]be concentric i.e. they have the common centre Centre of \[{{S}_{1}}\equiv \left( \frac{3}{2},\frac{-k}{2} \right),\]Centre of \[{{S}_{2}}\equiv \left( \frac{3}{2},\frac{1}{8} \right),\] \[\Rightarrow \frac{-k}{2}=\frac{1}{8}\Rightarrow k=\frac{-1}{4}\] Hence, option [c] is correct.

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