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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



The pair of tangents contain a right angle ,hence

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



Therefore,the locus of P is

x+a=0\, which is directrix.

Main Page:Geometry:The Parabola