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Find the length of the perpendicular from the origin on the line {\frac  {8}{r}}={\sqrt  {3}}\cos \theta +\sin \theta \,.Also determine the angle made by the perpendicular with the intial line.

Given line is {\frac  {8}{r}}={\sqrt  {3}}\cos \theta +\sin \theta \,.

The line in the normal form is p=r\cos(\theta -\alpha )\,

p=r\cos \theta \cos \alpha +r\sin \theta \sin \alpha \,

The above two equations represent the same line

{\frac  {p}{8}}={\frac  {\cos \alpha }{{\sqrt  {3}}}}={\frac  {\sin \alpha }{1}}\,

\cos \alpha ={\frac  {p{\sqrt  {3}}}{8}},\sin \alpha ={\frac  {p}{8}}\,

Squaring and adding

{\frac  {3p^{2}}{64}}+{\frac  {p^{2}}{64}}=1,p^{2}=16,p=4\,

Length of the perpendicular from origin on the given line is


\tan \alpha ={\frac  {\sin \alpha }{\cos \alpha }}={\frac  {1}{{\sqrt  {3}}}},\alpha =30^{\circ }\,

Main Page:Geometry:Polar Coordinates