Geo5.1.24

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Show that the locus of the foot of the perpendicular from the focus to the tangent of the parabola y^{2}=4ax\, is x=0\,,the tangent to the vertex.

Given parabola is y^{2}=4ax\,

Equation to any tangent to the parabola is y=mx+{\frac  {a}{m}}\,

Equation to the line perpendicular to the above line through S(a,0)\, is

m(x-a)+m^{2}y=0\,

x+my=a\,

Solving the two equations,we get

P(0,{\frac  {a}{m}})\, which is the foot of the perpendicular from S on the tangent.

Hence the locus of P is x=0\, which is the tangent at the vertex.


Main Page:Geometry:The Parabola