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JEE Main & Advanced Mathematics Conic Sections Question Bank Hyperbola

  • 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 [AIEEE 2005]

    A)            A parabola                                

    B)            A hyperbola

    C)            An ellipse                                  

    D)            A circle

    Correct Answer: B

    Solution :

               If \[y=mx+c\] is tangent to the hyperbola then \[{{c}^{2}}={{a}^{2}}{{m}^{2}}-{{b}^{2}}\]. Here \[{{\beta }^{2}}={{a}^{2}}{{\alpha }^{2}}-{{b}^{2}}\]. Hence locus of P(a, b) is \[{{a}^{2}}{{x}^{2}}-{{y}^{2}}={{b}^{2}}\], which is a hyperbola.


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