Geo4.132

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Tangents are drawn parallel to the line y=mx\, to touch the circles of the coaxal system x^{2}+y^{2}+2\lambda x+c=0\,.Show that the locus of their points of contact is the curve x^{2}+2mxy-y^{2}=c\,

Given coaxal system x^{2}+y^{2}+2\lambda x+c=0\,

Let P(x_1,y_1) be the point of contact. The equation of the tangent at(x1,y1) to the circle is

xx_{1}+yy_{1}+\lambda (x+x_{1})+c=0\,. equation be 2.

(x_{1}+\lambda )x+y_{1}y+(\lambda x_{1}+c)=0\,. this equation be 3.

P lies on the given equation

x_{1}^{{2}}+y_{1}^{{2}}+2\lambda x_{1}+c=0\,

Slope of 2 is

{\frac  {-(x_{1}+\lambda )}{y_{1}}}\,

Given that 2 is parallel to y=mx\,

Hence

{\frac  {-(x_{1}+\lambda )}{y_{1}}}=m\,

\lambda =-(x_{1}+my_{1})\,

Substituting the value of lambda in 3,we get

x_{1}^{{2}}+y_{1}^{{2}}-2x_{1}(x_{1}+my_{1})+c=0\,

x_{1}^{{2}}+2mx_{1}y_{1}-y_{1}^{{2}}-c=0\,

Therefore, locus of P is

x^{2}+2mxy-y^{2}-c=0\,


Main Page:Geometry:Circles