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Show that the locus of midpoints of the chord of the hyperbolax^{2}-y^{2}=a^{2}\,,which touch the parabola y^{2}=4ax\, is y^{2}(x-a)=x^{3}\,

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

The equation of the chord is xx_{1}-yy_{1}=x_{1}^{{2}}-y_{1}^{{2}}\,

This line touches the parabola y^{2}=4ax\,

The condition of tangency to the parabola is

c={\frac  {a}{m}}\,

In the above equations m={\frac  {x_{1}}{y_{1}}},c={\frac  {y_{1}^{{2}}-x_{1}^{{2}}}{y_{1}}}\,

Hence the condition is {\frac  {y_{1}^{{2}}-x_{1}^{{2}}}{y_{1}}}={\frac  {ay_{1}}{x_{1}}}\,

By simplifying the equation above,we get


Therefore,the locus is y^{2}(x-a)=x^{3}\,

Main Page:Geometry:Hyperbola