Geo5.3.30

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Tangents to the hyperbola {\frac  {x^{2}}{a^{2}}}-{\frac  {y^{2}}{b^{2}}}=1\, make angles \alpha ,\beta \, with the transverse axis. Find the locus of their point of intersection if \tan \alpha +\tan \beta =k\,

Equation of tangent to given hyperbola is y=mx\pm {\sqrt  {a^{2}m^{2}-b^{2}}}\,

Let the tangents intersect at P. Now,P lies on the above equation y_{1}=mx_{1}\pm {\sqrt  {a^{2}m^{2}-b^{2}}}\,

(y_{1}-mx_{1})^{2}=a^{2}m^{2}-b^{2}\,

m^{2}(x_{1}^{{2}}-a^{2})-2mx_{1}y_{1}+y_{1}^{{2}}+b^{2}=0\,

Let m1,m2 are the two roots of the quadratic equation and \tan \alpha =m_{1},m_{2}=\tan \beta \,

Given \tan \alpha +\tan \beta =k,m_{1}+m_{2}=k\,

k={\frac  {2x_{1}y_{1}}{x_{1}^{{2}}-a^{2}}}\,

Therefore,the locus of P is 2xy=k(x^{2}-a^{2})\,


Main Page:Geometry:Hyperbola