Geo5.2.22

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Find the coordiantes of the points on the ellipse x^{2}+3y^{2}=37\, at which the normal is parallel to the line 6x-5y=2\,

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

The normal to the ellipse is

{\frac  {a^{2}x}{x_{1}}}-{\frac  {b^{2}y}{y_{1}}}=a^{2}-b^{2},a>b\,

{\frac  {37x}{x_{1}}}-{\frac  {37y}{3y_{1}}}={\frac  {74}{3}}\,

3y_{1}x-x_{1}y-2x_{1}y_{1}=0\,

Since this is parallel to the given line

{\frac  {3y_{1}}{6}}={\frac  {-x_{1}}{-5}}={\frac  {2x_{1}y_{1}}{2}}\,

{\frac  {y_{1}}{2}}=x_{1}y_{1},{\frac  {x_{1}}{5}}=x_{1}y_{1}\,

x_{1}=2,y_{1}=5\,

Therefore the coordinates of the point are (2,5).


Main Page:Geometry:The Ellipse