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The polar of P w.r.t the parabola y^{2}=4ax\, touches the circle x^{2}+y^{2}=4a^{2}\,. Find the locus of P.

The polar of P(x_{1},y_{1})\, w.r.t the given parabola is


This will touch the circle x^{2}+y^{2}=4a^{2}\,, then the distance crom the centre(0,0)

is equal to the radius.

{\frac  {-2ax_{1}}{{\sqrt  {y_{1}^{{2}}+4a^{2}}}}}=2a\,


Hence the locus of P is x^{2}+y^{2}=4a^{2}\,

Main Page:Geometry:The Parabola