Geo5.2.40

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Show that the equation of the auxiliary circle of the ellipse {\frac  {x^{2}}{a^{2}}}+{\frac  {y^{2}}{b^{2}}}=1\, is x^{2}+y^{2}=a^{2}\,.

The equation to the ellipse is {\frac  {x^{2}}{a^{2}}}+{\frac  {y^{2}}{b^{2}}}=1\,

The equation to the tangent of the ellipse is y=mx\pm {\sqrt  {a^{2}m^{2}+b^{2}}},y-mx=\pm {\sqrt  {a^{2}m^{2}+b^{2}}}\,

The equation to the perpendicular from either focus (\pm ae,0)\, on this tangent is given by y={\frac  {-1}{m}}(x\pm ae),my+x=\pm ae\,

To find the locus of point of intersection of the above two equations,eliminate m from them.

Squaring and adding the two equations,we get

(y-mx)^{2}+(my+x)^{2}=a^{2}m^{2}+b^{2}+a^{2}e^{2}\,

x^{2}(1+m^{2})+y^{2}(1+m^{2})=a^{2}m^{2}+a^{2}(1-e^{2})+a^{2}e^{2}\,

(x^{2}+y^{2})(1+m^{2})=a^{2}(m^{2}+1)\,

x^{2}+y^{2}=a^{2}\,

Therefore the locus is the auxiliary circle concentric with the ellipse.


Main Page:Geometry:The Ellipse