Geo4.68

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Tangents are drawn to the circle x^{2}+y^{2}=a^{2}\, from a point which always lies on the line lx+my=1\,.Prove that the locus of the mid-point of the chords of contact is x^{2}+y^{2}-a^{2}(lx+my)=0\,.

Given circle is x^{2}+y^{2}=a^{2}\,

Let this be 1.

Given line is lx+my=1\,

Let this equation be 2.

Let (h,k) be a point on the line 2, hence lh+mk=1\,.

This is equation 3

The equation of the chord of contact of tangents from (h,k)to the circle is hx+ky=a^{2}\,

Let this be equation 4

Let (x1,y1) be the mid point of the chord, then teh equation of the chord is

xx_{1}+yy_{1}=x_{1}^{{2}}+y_{1}^{{2}}\,

Le this be 5.

The equations 4 and 5 represent the same.

Hence

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

Therefore h={\frac  {a^{2}x_{1}}{x_{1}^{{2}}+y_{1}^{{2}}}},k={\frac  {a^{2}y_{1}}{x_{1}^{{2}}+y_{1}^{{2}}}}\,

Substituting these values in the equation 3, we get

{\frac  {la^{2}x_{1}}{x_{1}^{{2}}+y_{1}^{{2}}}}+{\frac  {ma^{2}y_{1}}{x_{1}^{{2}}+y_{1}^{{2}}}}=1\,

a^{2}(lx_{1}+my_{1})=x_{1}^{{2}}+y_{1}^{{2}}\,

Hence the locus of (x1,y1) is x^{2}+y^{2}-a^{2}(lx+my)=0\,


Main Page:Geometry:Circles