Geo4.63

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Find the locus of the midpoints of chords of the circle x^{2}+y^{2}=r^{2}\,,subtending a right angle at the point (a,b).

Given circle is

S=x^{2}+y^{2}-r^{2}=0\,

Let P(x,y) be the mid point of a chord AB of the circle S=0.

P(x,y) lies inside S=0

S_{{11}}<0\,

|S_{{11}}|=-S_{{11}}\,

Length of the chord AB is

2{\sqrt  {|S_{{11}}|}}\,

AP={\sqrt  {|S_{{11}}|}},[AP]^{2}=|S_{{11}}|=-S_{{11}}\,

Let C=(a,b). Given \angle ACB=\ 90^{\circ }\,

Circle on AB as diameter passes through C.

PC=PA\,

[PC]^{2}=[PA]^{2}\,

(x_{1}-a)^{2}+(y_{1}-b)^{2}=-S_{{11}}\,

x_{1}^{{2}}-2ax_{1}+a^{2}+y_{1}^{{2}}-2by_{1}+b^{2}=-(x_{1}^{{2}}+y_{1}^{{2}}-r^{2})\,

Therefore,locus of P is

2x^{2}+2y^{2}-2ax-2by+a^{2}+b^{2}-r^{2}=0\,

Main Page:Geometry:Circles