Geo5.3.42

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Find the locus of midpoints of the chords of hyperbola {\frac  {x^{2}}{a^{2}}}-{\frac  {y^{2}}{b^{2}}}=1\, drawn parallel to the line y=mx\,

Let P be the midpoint of the chord

Equation of the chord is {\frac  {xx_{1}}{a^{2}}}-{\frac  {yy_{1}}{b^{2}}}={\frac  {x_{1}^{{2}}}{a^{2}}}-{\frac  {y_{1}^{{2}}}{b^{2}}}\,

This line is paralell to the line y=mx\,

Hence the slopes of the two equations are equal,therefore,

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

b^{2}x_{1}=ma^{2}y_{1}\,

Hence the locus of P is

ma^{2}y-b^{2}x=0\,


Main Page:Geometry:Hyperbola