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Prove that the product of the perpendiculars from any point on the hyperbola{\frac  {x^{2}}{a^{2}}}-{\frac  {y^{2}}{b^{2}}}=1\, to its asymptotes is constant.

Let P(a\sec \theta ,b\tan \theta )\, be any point on the hyperbola {\frac  {x^{2}}{a^{2}}}-{\frac  {y^{2}}{b^{2}}}=1\,

The asymptotes are {\frac  {x}{a}}+{\frac  {y}{b}}=0,{\frac  {x}{a}}-{\frac  {y}{b}}=0\,

Product of the perpendiculars from P to the aymptotes is

|{\frac  {\sec \theta +\tan \theta }{{\sqrt  {{\frac  {1}{a^{2}}}+{\frac  {1}{b^{2}}}}}}}||{\frac  {\sec \theta -\tan \theta }{{\sqrt  {{\frac  {1}{a^{2}}}+{\frac  {1}{b^{2}}}}}}}|\,

{\frac  {\sec ^{2}\theta -\tan ^{2}\theta }{{\frac  {1}{a^{2}}}+{\frac  {1}{b^{2}}}}}\,

{\frac  {1}{{\frac  {1}{a^{2}}}+{\frac  {1}{b^{2}}}}}={\frac  {a^{2}b^{2}}{a^{2}+b^{2}}}\, which is a constant.

Main Page:Geometry:Hyperbola