Geo5.3.39

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Show that the locus of midpoints of the chord of the hyperbola{\frac  {x^{2}}{a^{2}}}-{\frac  {y^{2}}{b^{2}}}=1\,,which pass through the focus (ae,0)\, is {\frac  {x^{2}}{a^{2}}}-{\frac  {y^{2}}{b^{2}}}={\frac  {ex}{a}}\,

Let the mid point be P(x_{1},y_{1})\, of the chord

The equation to the chord is

b^{2}xx_{1}-a^{2}yy_{1}=b^{2}x_{1}^{{2}}-a^{2}y_{1}^{{2}}\,

This equation pass through (ae,0)\,

b^{2}aex_{1}=b^{2}(ae)^{2}-a^{2}y_{1}^{{2}}\,


By simplifying the equation,we get

{\frac  {x_{1}^{{2}}}{a^{2}}}-{\frac  {y_{1}^{{2}}}{b^{2}}}={\frac  {ex_{1}}{a}}\,

Therefore,the locus of P is

{\frac  {x^{2}}{a^{2}}}-{\frac  {y^{2}}{b^{2}}}={\frac  {ex}{a}}\,


Main Page:Geometry:Hyperbola