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KVPY Sample Paper KVPY Stream-SX Model Paper-18

  • question_answer
    Let P \[\left( 3\sec \theta ,2\tan \theta  \right)\] and Q \[\left( 3\sec \phi ,2\tan \phi  \right)\] where \[\theta +\phi =\frac{\pi }{2},\] be two distinct points on the hyperbola \[\frac{{{x}^{2}}}{9}-\frac{{{y}^{2}}}{4}=1.\]then the ordinate of the point of intersection of the normal at P and Q is:

    A) \[\frac{11}{3}\]

    B) \[-\frac{11}{3}\]

    C) \[\frac{13}{2}\]

    D)  \[-\frac{13}{2}\]

    Correct Answer: D

    Solution :

    let the coordinate at point of intersection of normal at P and Q be \[\left( h,k \right)\]
    Since, equation of normal to the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] at point \[\left( {{x}_{1}},{{y}_{1}} \right)\] is \[\frac{{{a}^{2}}x}{{{x}_{1}}}+\frac{{{b}^{2}}y}{{{y}_{1}}}\]\[={{a}^{2}}+{{b}^{2}}\] therefore equation of normal to the hyperbola \[\frac{{{x}^{2}}}{{{3}^{2}}}-\frac{{{y}^{2}}}{{{2}^{2}}}=1\] at point \[\operatorname{P}\left( 3\sec \operatorname{q},2tan\theta  \right)\] is
    \[\frac{{{3}^{2}}x}{3\sec \theta }+\frac{{{2}^{2}}y}{2\tan \theta }={{3}^{2}}+{{2}^{2}}\]\[\Rightarrow 3x\cos \theta +2y\cot \theta ={{3}^{2}}+{{2}^{2}}..(1)\]
    Similarly, equation of normal to the hyperbola \[\frac{{{x}^{2}}}{{{3}^{2}}}-\frac{{{y}^{2}}}{{{2}^{2}}}\]at point \[\operatorname{Q}\left( 3\sec \phi ,2tan\phi  \right)\]is \[\frac{{{3}^{2}}x}{3\sec \phi }+\frac{{{2}^{2}}y}{2\tan \phi }={{3}^{2}}+{{2}^{2}}\]\[\Rightarrow 3x\cos \phi +2y\cot \phi ={{3}^{2}}+{{2}^{2}}..(2)\]
    Given \[\theta +\phi =\frac{\pi }{2}\Rightarrow \phi =\frac{\pi }{2}-\theta \] and these passes through \[\left( h,k \right)\]
    \[\therefore \]from eq. (2) \[3x\cos \left( \frac{\pi }{2}-\theta  \right)+2y\cot \left( \frac{\pi }{2}-\theta  \right)={{3}^{2}}+{{2}^{2}}\]\[\Rightarrow 3h\cos \theta +2k\cot \theta ={{3}^{2}}+{{2}^{2}}..(3)\]
    and \[3h\cos \theta +2k\cot \theta ={{3}^{2}}+{{2}^{2}}\,\,\,\,\,\,\,\,..(4)\]
    Comparing equation (3) & (4), we get \[3h\cos \theta +2k\cot \theta =3h\sin \theta +2k\tan \theta \]
    \[3h\cos \theta -3h\sin \theta =2k\tan \theta -2k\cot \theta \]
    \[3h\left( \cos \theta -\sin \theta  \right)=2k\left( \tan \theta -\cot \theta  \right)\]
    \[3h\left( \cos \theta -\sin \theta  \right)\]\[=2k\frac{\left( \sin \theta -\cos \theta  \right)\left( \sin \theta +\cos \theta  \right)}{\sin \theta \cos \theta }\]
    \[\operatorname{or}\,\,\,3h=\frac{-2k\left( \sin \theta +\cos \theta  \right)}{\sin \theta \cos \theta }..(5)\]
    Now, putting the value of equation (5) in eq. (3) \[\frac{-2k\left( \sin \theta +\cos \theta  \right)\sin \theta }{\sin \theta \cos \theta }+2k\tan \theta ={{3}^{2}}+{{2}^{2}}\]\[\Rightarrow 2k\tan \theta -2k+2k\tan \theta =13\]\[-2k=13\Rightarrow k=\frac{-13}{2}\] Hence, ordinate of point of intersection of normal at P and Q is \[\frac{-13}{2}\]


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