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# Determine the equation of family of circles passing through two given points A(x1,y1) and B(x2,y2) . If fixed circle S1 cuts each member of the family in various chords then prove that all these chords pass through a fixed point.

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## We can determine the equation of a circle passing through three points as its equation involves three unknowns g, f and c. However, if only two points A and B are given, we can find the family of circles given byS + λL = 0 .(1)where S is circle on AB as diameter and L is the line AB.S1 = 0 ..(2)is another fixed circle.Common chords of (1) and (2) are given byRule: S1S2 = 0 i.e., (S + λL)S1 = 0or (S S1)+λ L = 0 or U + λL = 0,where U is common chord of S and S1 which is fixed. But U + λL = 0 represents a family of straight lines (i.e., common chords) which pass through the intersection of fixed lines U = 0 and L = 0 and hence a fixed point.  Suggest Corrections  0      Similar questions  Related Videos   Parabola
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