11th Class Mathematics Conic Sections Question Bank Critical Thinking

  • question_answer The locus of the midpoint of the line segment joining the focus to a moving point on the parabola \[{{y}^{2}}=4ax\] is another parabola with the directrix                                                                                   [IIT Screening 2002]

    A)            \[x=-a\]                                     

    B)            \[x=-\frac{a}{2}\]

    C)            \[x=0\]                                      

    D)            \[x=\frac{a}{2}\]

    Correct Answer: C

    Solution :

               \[\alpha =\frac{a{{t}^{2}}+a}{2},\,\,\beta =\frac{2at+0}{2}\Rightarrow \,2\alpha =a{{t}^{2}}+a,\,at=\beta \]                   \ \[2\alpha =a.\,\frac{{{\beta }^{2}}}{{{a}^{2}}}+a\] or \[2a\alpha ={{\beta }^{2}}+{{a}^{2}}\]                   \ The locus is \[{{y}^{2}}=\frac{4a}{2}\left( x-\frac{a}{2} \right)\] \[=4b(x-b),\,\left( b=\frac{a}{2} \right)\]                    \ Directrix is \[(x-b)\,+b=0\] or \[x=0\].

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