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JEE Main & Advanced Mathematics Conic Sections Question Bank Self Evaluation Test - Conic Sections

  • question_answer
                Let \[P(a\,\,sec\theta ,b\,\,tan\,\theta )\] and Q \[Q(a\,\,sec\,\phi ,\,\,b\,tan\,\,\phi ),\]where \[\theta +\phi =\pi /2,\] be two points on the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1.\] If \[(h,k)\] is the point of intersection of the normal at P and Q, then kz is equal to

    A) \[\frac{{{a}^{2}}+{{b}^{2}}}{a}\]

    B) \[-\left( \frac{{{a}^{2}}+{{b}^{2}}}{a} \right)\]

    C) \[\frac{{{a}^{2}}+{{b}^{2}}}{b}\]

    D) \[-\left( \frac{{{a}^{2}}+{{b}^{2}}}{b} \right)\]

    Correct Answer: D

    Solution :

    [d] Equation of the normal to the hyperbola\[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] at the point \[(aseca,btana)\] is given by \[ax\cos \alpha +by\cot \alpha ={{a}^{2}}+{{b}^{2}}\] Normal at \[\theta ,\phi \] are \[\left\{ \begin{matrix}    ax\cos \theta +by\cot \theta ={{a}^{2}}+{{b}^{2}}  \\    ax\cos \phi +by\cot \phi ={{a}^{2}}+{{b}^{2}}  \\ \end{matrix} \right.\] where \[\phi =\frac{\pi }{2}-\theta \] and these passes through \[(h,k)\] \[\therefore \,\,\,\,ah\cos \theta +bk\cot \theta ={{a}^{2}}+{{b}^{2}}\] \[\Rightarrow ah\sin \theta +bk\tan \theta ={{a}^{2}}+{{b}^{2}}\] Eliminating \[h,\,\,bk(cot\theta sin\theta -tan\theta cos\theta )\] \[=({{a}^{2}}+{{b}^{2}})(sin\theta -cos\theta )\] or \[k=-({{a}^{2}}+{{b}^{2}})/b\]


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