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JEE Main & Advanced Sample Paper JEE Main Sample Paper-16

  • question_answer
    A variable line having intercepts e and e' on co-ordinate axes, where\[\frac{e}{2},\frac{e'}{2}\] are eccentricities of a hyperbola and its conjugate hyperbola, then the line touches the circle \[{{x}^{2}}+{{y}^{2}}={{r}^{2}}\] whose radius is or      

    A)  1                                            

    B)  2

    C)  3                                            

    D)  cannot be found

    Correct Answer: B

    Solution :

    Since\[\frac{e}{2}\]and \[\frac{e'}{2}\]are eccentricities of hyperbola and its conjugate hyperbola hence,\[\frac{4}{{{e}^{2}}}+\frac{4}{e{{'}^{2}}}=1\]                                                         ?(1) equation of the line is \[\frac{x}{e}+\frac{y}{e'}=1\] \[e'x+ey-ee'=0\]which touches the circle \[{{x}^{2}}+{{y}^{2}}={{r}^{2}}\]Hence,\[\frac{ee'}{\sqrt{{{e}^{2}}+{{(e')}^{2}}}}=r\Rightarrow {{r}^{2}}=4\]or\[r=2\{\text{from}\,(1)\}\]


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