Geo5.2.41

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Tangents at right angles are drawn to the ellipseS=0\,.Show that the locus of the midpoints of chords of contact is the curve [{\frac  {x^{2}}{a^{2}}}+{\frac  {y^{2}}{b^{2}}}]^{2}={\frac  {x^{2}+y^{2}}{a^{2}+b^{2}}}\,

Let P(x_{1},y_{1})\, be the mid point of a chord QR of the given ellipse.

Equation to chord QR is S_{1}=S_{{11}}\,

{\frac  {xx_{1}}{a^{2}}}+{\frac  {yy_{1}}{b^{2}}}={\frac  {x_{1}^{{2}}}{a^{2}}}+{\frac  {y_{1}^{{2}}}{b^{2}}}=k\, (say)

Let the tangents at Q and R to the ellipse meet at M.

Therefore coordinates of M are \left({\frac  {-a^{2}x_{1}}{a^{2}k}},{\frac  {-b^{2}y_{1}}{b^{2}k}}\right)\,

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

Given that the tangents MQ and MR are perpendiculars.

Mlies on director circle x^{2}+y^{2}=a^{2}+b^{2}\,

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

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

Therefore the locus of P is

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


Main Page:Geometry:The Ellipse