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JEE Main & Advanced Mathematics Conic Sections Question Bank Mock Test - Conic Sections

  • question_answer
    The locus of a point P(\[\alpha \],\[\beta \])moving under the condition that the line \[y=\alpha x+\beta \] is a tangent to the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\]is

    A) An ellipse        

    B) A circle

    C) A parabola     

    D) A hyperbola

    Correct Answer: D

    Solution :

    [d] The tangent to the hyperbola\[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] is \[y=mx\pm \sqrt{{{a}^{2}}{{m}^{2}}-{{b}^{2}}}\] Given that \[y=\alpha x+\beta \] is a tangent of the hyperbola. So \[m=\alpha \]and \[{{a}^{2}}{{m}^{2}}-{{b}^{2}}={{\beta }^{2}}\] \[\therefore {{a}^{2}}{{\alpha }^{2}}-{{b}^{2}}={{\beta }^{2}}\] The locus is \[{{a}^{2}}{{x}^{2}}-{{y}^{2}}={{b}^{2,}}\]which is a hyperbola.


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