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A tangent of the auxiliary circle of the hyperbola {\frac  {x^{2}}{a^{2}}}-{\frac  {y^{2}}{b^{2}}}=1\, intersects it in P and Q.Find the locus of midpoint of PQ.

Let R(x_{1},y_{1})\, be the midpoint of a chord PQ of the given hyperbola

Equation to chord of PQ is {\frac  {xx_{1}}{a^{2}}}-{\frac  {yy_{1}}{b^{2}}}={\frac  {x_{1}^{{2}}}{a^{2}}}-{\frac  {y_{1}^{{2}}}{b^{2}}}\,

It is a tangent to the auxiliary circle x^{2}+y^{2}=a^{2}\,


{\frac  {|{\frac  {x_{1}^{{2}}}{a^{2}}}-{\frac  {y^{{2}}}{b^{2}}}|}{{\sqrt  {{\frac  {x_{1}^{{2}}}{a^{4}}}+{\frac  {y_{1}^{{2}}}{b^{4}}}}}}}=a\,

Therefore,the locus of R is [{\frac  {x^{2}}{a^{2}}}-{\frac  {y^{2}}{b^{2}}}]^{2}=a^{2}[{\frac  {x^{2}}{a^{4}}}+{\frac  {y^{2}}{b^{4}}}]\,

Main Page:Geometry:Hyperbola