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JEE Main & Advanced Mathematics Circle and System of Circles Question Bank Tangent and normal to a circle

  • question_answer
    The line \[x\cos \alpha +y\sin \alpha =p\]will be a tangent to the circle \[{{x}^{2}}+{{y}^{2}}-2ax\cos \alpha -2ay\sin \alpha =0\], if \[p=\]

    A)            0 or a                                        

    B)            0

    C)            \[2a\]                                         

    D)            \[0\]or \[2a\]

    Correct Answer: D

    Solution :

               \[x\cos \alpha +y\sin \alpha -p=0\] is a tangent, if perpendicular from centre on it is equal to radius of the circle. Here centre is \[(a\cos \alpha ,\ a\sin \alpha )\] and radius is a.                    \ \[\left| \frac{a{{\cos }^{2}}\alpha +a{{\sin }^{2}}\alpha -p}{\sqrt{1}} \right|=a\]                    i.e. \[|a-p|\ =a\Rightarrow p=0\] or \[p=2a\].


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