A circle S, cuts a rectangular hyperbola xy = c 2 in points A, B, C, D whose parameters are t 1 , t 2 , t 3 and t 4 respectively
(i) The value of t 1 t 2 t 3 t 4 using the above conditions are
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Ans.
(i)
Sol. Circle x 2 + y 2 + 2gx + 2fy + c = 0
hyperbola x = ct, y = 
; t ∈ R 0
c 2 t 2 + 
+ 2g.ct + 2f 
+ k = 0
c 2 t 4 + 2gct 3 + kt 2 + 2fct + c 2 = 0
t 1 t 2 t 3 t 4 = 1
(ii)
Sol. Centre of mean of points of intersection of circle and hyperbola lies at the mid point of line joining their centre
Let (h, k) be the centre of circle 
= h & k = 
(iii)
Sol. Let centroid be ( α, β )
x 1 = 2h – 3 α & y 1 = 2k – 3 β and (x 1 ,y 1 ) lie in hyperbola
xy = c 2 locus of ( α , β ) is also a hyperbola.
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