Geo4.23

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Show that the pair of straight lines ax^{2}+2hxy+ay^{2}+2gx+2fy+c=0\, meet the coordinate axes in concyclic points.Also find the equation of the circle through those cyclic points.

Given equation to the pair of straight lines is

ax^{2}+2hxy+ay^{2}+2gx+2fy+c=0\,

Let the lines be

l_{1}x+m_{1}y+n_{1}=0,l_{2}x+m_{2}y+n_{2}=0\,

Comparing the coefficients,

l_{1}l_{2}=a,l_{1}m_{2}+l_{2}m_{1}=2h,m_{1}m_{2}=a\,

Now

l_{1}l_{2}=m_{1}m_{2}=a\,

The lines cut the axes in concyclic points.

Now the equation to the circle is

(l_{1}x+m_{y}+n_{1})(l_{2}x+m_{2}y+n_{2})-(l_{1}m_{2}+l_{2}m_{1})xy=0\,

ax^{2}+2hxy+ay^{2}+2gx+2fy+c-2hxy=0\,

ax^{2}+ay^{2}+2gx+2fy+c=0\,


Main Page:Geometry:Circles