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JAMIA MILLIA ISLAMIA Jamia Millia Islamia Solved Paper-2007

  • question_answer
        If\[a\ne 0\]and the line\[2bx+3cy+4d=0\]passes through the points of intersection of the parabolas\[{{y}^{2}}=4ax\]and\[{{x}^{2}}=4ay,\]then

    A)  \[{{d}^{2}}+{{(2b+3c)}^{2}}=0\]

    B)  \[{{d}^{2}}+{{(3b+2c)}^{2}}=0\]

    C)  \[{{d}^{2}}+{{(2b-3c)}^{2}}=0\]

    D)  \[{{d}^{2}}+{{(3b-2c)}^{2}}=0\]

    Correct Answer: A

    Solution :

                    The equation of parabolas are \[{{y}^{2}}=4ax\] and\[{{x}^{2}}=4ay\]. On solving these, we get \[x=0\] and\[x=4a\] Also\[y=0\]and\[y=4a.\] \[\therefore \]The point of intersection of parabolas are A(0,0) and\[B(4a,4a)\]. Also line\[2bx+3cy+4d=0\]passes through A and B. \[\therefore \]                  \[d=0\]                                 ...(i) and        \[2b.4a+3c-4a+4d=0\] \[\Rightarrow \]               \[2ab+3ac+d=0\] \[\Rightarrow \]          \[a(2b+3c)=0\]                               \[(\because d=0)\] \[\Rightarrow \]            \[2b+3c=0\]                                        ...(ii) On squaring Eqs. (i) and (ii) and then adding, we get \[{{d}^{2}}+{{(2b+3c)}^{2}}=0\]


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