Geo5.3.25

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Show that the locus of the poles w.r.t the parabola y^{2}=4ax\, of tangents to the hyperbola 4x^{2}-3y^{2}=a^{2}\, is 12x^{2}+y^{2}=3a^{2}\,

Polar of the pole P w.r.t the given parabola is yy_{1}-2ax-2ax_{1}=0\,

This is a tangent to the hyperbola {\frac  {x^{2}}{{\frac  {a^{2}}{4}}}}-{\frac  {y^{2}}{{\frac  {a^{2}}{3}}}}=1\,

The condition of tangency is a^{2}l^{2}-b^{2}m^{2}=n^{2}\,

Therefore,

{\frac  {a^{2}}{4}}(4a^{2})-{\frac  {a^{2}}{3}}(y_{1}^{{2}})=4a^{2}x_{1}^{{2}}\,

3a^{4}-a^{2}y_{1}^{{2}}=12a^{2}x_{1}^{{2}}\,

12x_{1}^{{2}}+y_{1}^{{2}}=3a^{2}\,

Therefore,the locus of P is

12x^{2}+y^{2}=3a^{2}\,

Main Page:Geometry:Hyperbola