Geo5.1.22

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Show that the locus of the point of intersection of perpendicular tangents to the parabola y^{2}=4ax\, is the directrix x+a=0\,

Let two tangents be drawn from P(x_{1},y_{1})\, to the given parabola

Therefore equation to pair of tangents from P is

[S_{1}]^{2}=SS_{{11}}\,

[yy_{1}-2a(x+x_{1})^{2}]=(y^{2}-4ax)(y_{1}^{{2}}-4ax_{1})\,

The pair of tangents contain a right angle ,hence

Ceofficient of x^{2}\,+coefficient of y^{2}\,=0\,

4a^{2}+y_{1}^{{2}}-(y_{1}^{{2}}-4ax_{1})=0\,

4a^{2}+4ax_{1}=0\,

Therefore,the locus of P is

x+a=0\, which is directrix.


Main Page:Geometry:The Parabola