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Show that the locus of poles w.r.t hyperbola {\frac  {x^{2}}{a^{2}}}-{\frac  {y^{2}}{b^{2}}}=1\, of tangents to the parabola y^{2}=4ax\, is a^{3}y^{2}+b^{4}x=0\,

Given parabola is {\frac  {x^{2}}{a^{2}}}-{\frac  {y^{2}}{b^{2}}}=1\,

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

{\frac  {xx_{1}}{a^{2}}}-{\frac  {yy_{1}}{b^{2}}}=1\,

It will be a tangent to y^{2}=4ax\, if {\frac  {x_{1}}{a^{2}}}(-1)=a[{\frac  {-y_{1}}{b^{2}}}]^{2}\,.

Therefore,the locus of P is a^{3}y^{2}+b^{4}x=0\,

Main Page:Geometry:Hyperbola