Geo5.2.38

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A chord PQ of an ellipse subtends a right angle at the centre of the ellipse S=0\,.Show that the point of intersection of tangents at P and Q lies on the ellipse{\frac  {x^{2}}{a^{4}}}+{\frac  {y^{2}}{b^{4}}}={\frac  {1}{a^{2}}}+{\frac  {1}{b^{2}}}\,

Let the tangents at P and Q intersects at R(x_{1},y_{1})\,,then PQ is the chord of tangents drawn from (x_{1},y_{1})\,

Therefore PQ={\frac  {xx_{1}}{a^{2}}}+{\frac  {yy_{1}}{b^{2}}}=1\,

The equation of the pair of lines CP and CQ is obtained by homogenising the equation of S=0 with PQ.

The combined equation is {\frac  {x^{2}}{a^{2}}}+{\frac  {y^{2}}{b^{2}}}=[{\frac  {xx_{1}}{a^{2}}}+{\frac  {yy_{1}}{b^{2}}}]^{2}\,

x^{2}[{\frac  {1}{a^{2}}}-{\frac  {x_{1}^{{2}}}{a^{4}}}]-{\frac  {2x_{1}y_{1}xy}{a^{2}b^{2}}}+y^{2}[{\frac  {1}{b^{2}}}-{\frac  {y_{1}^{{2}}}{b^{4}}}]=0\,

But angle PCQ is 90.

Therefore,the coefficients of square of x and square of y is equal to zero.

[{\frac  {1}{a^{2}}}-{\frac  {x_{1}^{{2}}}{a^{4}}}]+[{\frac  {1}{b^{2}}}-{\frac  {y_{1}^{{2}}}{b^{4}}}]=0\,

{\frac  {x_{1}^{{2}}}{a^{4}}}+{\frac  {y_{1}^{{2}}}{b^{4}}}={\frac  {1}{a^{2}}}+{\frac  {1}{b^{2}}}\,

Hence the point of intersection of tangents at P and Q lies on the ellipse

{\frac  {x^{2}}{a^{4}}}+{\frac  {y^{2}}{b^{4}}}={\frac  {1}{a^{2}}}+{\frac  {1}{b^{2}}}\,

Main Page:Geometry:The Ellipse