Geo5.1.44

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Prove that the circle on a focal radius of a prabola,as diameter touches the tangent at the vertex.

Let the parabola be y^{2}=4ax\,.

Let P(at^{2},2at)\, be a point on the parabola and S=(a,0).

Equation to the circle on SP as diameter

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

x^{2}+y^{2}-ax(1+t^{2})-2aty+a^{2}t^{2}=0\,

In this circle f=-at,c=a^{2}t^{2}\,

and f^{2}-c=a^{2}t^{2}-a^{2}t^{2}=0\,

Circle touches the y-axis.

Hence the circle touches the tangent at the vertex.


Main Page:Geometry:The Parabola