Geo5.2.50

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Show that the locus of poles of chords of ellipse S=0\, which touch the parabola y^{2}=4px\, is pa^{2}y^{2}+b^{4}x=0\,

Given ellipse is {\frac  {x^{2}}{a^{2}}}+{\frac  {y^{2}}{b^{2}}}=1\,

Let P(x_{1},y_{1})\, be the pole of a chord of ellipse.

Therefore,equation of polar w.r.t the ellipse given is

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

This line touches the parabola y^{2}=4px\,

Hence

{\frac  {x_{1}}{a^{2}}}(-1)=p[{\frac  {y_{1}}{b^{2}}}]^{2}\,

pa^{2}y_{1}^{{2}}+b^{4}x_{1}=0\,

Therefore,locus of P is

pa^{2}y^{2}+b^{4}x=0\,


Main Page:Geometry:The Ellipse