Geo4.130

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Find the equation of a circle which passes through the origin and belongs to the coaxal system of which (1,2) (4,3)are the limiting points.

Equations of limiting circles are

(x-1)^{2}+(y-2)^{2}=0,(x-4)^{2}+(y-3)^{2}=0\,

x^{2}+y^{2}-2x-4y+5=0,x^{2}+y^{2}-8x-6y+25=0\,

Equation to the radical axis is

3x+y-10=0\,

Any circle coaxal with the above circles is

x^{2}+y^{2}-2x-4y+5+k(3x+y-10)=0\,

This will pass through origin if 5-10k=0,k={\frac  {1}{2}}\,

Therefore the required circle is

2(x^{2}+y^{2})-x-7y=0\,


Main Page:Geometry:Circles