Geo5.3.6

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A hyperbola has one focus at the origin and its eccentricity is {\sqrt  {2}}\,.One of its directories is x+y+1=0\,.Find the equation of the hyperbola.

Given S=(0,0),e={\sqrt  {2}},l\equiv x+y+1=0\,

Let P(x_{1},y_{1})\, be any point on the parabola and PM be the perpendicular on l.

By definition SP^{{2}}=e^{2}\cdot PM^{{2}}\,

(x_{1}-0)^{2}+(y_{1}-0)^{2}=2[{\frac  {x_{1}+y_{1}+1}{{\sqrt  {2}}}}]\,

x_{1}^{{2}}+y_{1}^{{2}}=(x_{1}+y_{1}+1)^{2}\,

x_{1}^{{2}}+y_{1}^{{2}}=x_{1}^{{2}}+y_{1}^{{2}}+1+2x_{1}y_{1}+2x_{1}+2y_{1}\,

2x_{1}y_{1}+2x_{1}+2y_{1}+1=0\,

Therefore,locus of the point is 2xy+2x+2y+1=0\, which is the equation of the hyperbola.


Main Page:Geometry:Hyperbola