Geo5.3.17

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Show that the line x\cos \alpha +y\sin \alpha =p\, touches the hyperbola {\frac  {x^{2}}{a^{2}}}-{\frac  {y^{2}}{b^{2}}}=1\, if a^{2}\cos ^{2}\alpha -b^{2}\sin ^{2}\alpha =p^{2}\,

From the given line and hyperbola

l=\cos \alpha ,m=\sin \alpha ,n=p\,

The condition of tangency is a^{2}l^{2}-b^{2}m^{2}=n^{2}\,

Substituting the values in the condition, we get

a^{2}\cos ^{2}\alpha -b^{2}\sin ^{2}\alpha =p^{2}\,

Hence the required.


Main Page:Geometry:Hyperbola