11th Class Mathematics Conic Sections Question Bank Critical Thinking

  • question_answer Tangent is drawn to ellipse \[\frac{{{x}^{2}}}{27}+{{y}^{2}}=1\] at \[(3\sqrt{3}\cos \theta ,\ \sin \theta )\] where \[\theta \in (0,\ \pi /2)\]. Then the value of \[\theta \] such that sum of intercepts on axes made by this tangent is minimum, is         [IIT Screening 2003]

    A)            \[\pi /3\]                                   

    B)            \[\pi /6\]

    C)            \[\pi /8\]                                   

    D)            \[\pi /4\]

    Correct Answer: B

    Solution :

               \[\frac{x\cos \theta }{3\sqrt{3}}+y\sin \theta =1.\]            Sum of intercepts = \[3\sqrt{3}\]\[\sec \theta +\text{cosec}\,\theta =f(\theta )\], (say)            \[f'\,(\theta )=\frac{3\sqrt{3}{{\sin }^{3}}\theta -{{\cos }^{3}}\theta }{{{\sin }^{2}}\theta \,{{\cos }^{2}}\theta }\]. At \[\theta =\frac{\pi }{6},\,f(\theta )\] is minimum.

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