Geo5.2.42

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Find the locus of the midpoints of chords of an ellipse,whose poles lie on the auxiliary circle.

Let the ellipse be {\frac  {x^{2}}{a^{2}}}+{\frac  {y^{2}}{b^{2}}}=1\,

Let P(x_{1},y_{1})\, be the midpoint of a chord AB of the ellipse.

Therefore,the equation to AB is {\frac  {xx_{1}}{a^{2}}}+{\frac  {yy_{1}}{b^{2}}}={\frac  {x_{1}^{{2}}}{a^{2}}}+{\frac  {y_{1}^{{2}}}{b^{2}}}=k\, (say)

The pole of the above line w.r.t the ellipse is

Q\left({\frac  {-x_{1}}{k}},{\frac  {-y_{1}}{k}}\right)\,

But Q lies on auxiliary circle x^{2}+y^{2}=a^{2}\,

Hence, x_{1}^{{2}}+y_{1}^{{2}}=k^{2}a^{2}\,

x_{1}^{{2}}+y_{1}^{{2}}=a^{2}[{\frac  {x_{1}^{{2}}}{a^{2}}}+{\frac  {y_{1}^{{2}}}{b^{2}}}]^{2}\,

Therefore,the locus of P is

x^{2}+y^{2}=a^{2}[{\frac  {x^{2}}{a^{2}}}+{\frac  {y^{2}}{b^{2}}}]^{2}\,


Main Page:Geometry:The Ellipse