KVPY Sample Paper KVPY Stream-SX Model Paper-19

  • question_answer
    A variable chord PQ of parabola \[{{y}^{2}}=4ax\] subtends a right angle at the vertex. Find the locus of point of intersection of the tangents at P and Q.

    A) 0

    B) 1    

    C) 2         

    D) none

    Correct Answer: A

    Solution :

    Let \[y=mx+c\] is the variable chord which is subtending right angle at the vertex of parabola hence equation of pair of straight lines OP and OQ can be given by making a homogeneous second degree equation with the help of parabola and chord as follows
    \[{{y}^{2}}-4ax\left( \frac{y-mx}{c} \right)=0\]
    for subtend \[90{}^\circ \] at vertex \[1+\frac{4am}{c}=0\]
    \[\Rightarrow c=-\,4am\]\[\Rightarrow \]equation of chord is \[y=m\,\,(x-4a)\]
    Let point of intersection of tangents at the extremeities is \[({{x}_{1}},{{y}_{1}})\]
    Which is passing through \[(4a,0)\]
    \[\Rightarrow {{x}_{1}}+4a=0\]\[\Rightarrow x+4a=0\] is the required locus.

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