Circles are drawn on chords of rectangular hyperbola xy = c 2 parallel to the line y = x as diameters. All such circles pass through two fixed points whose co-ordinates are :
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(a,d)Equation of chord joining 
and 
is
x + t 1 t 2 y = (t 1 + t 2 ) c ........(1)
(i) is parallel to y = x
⇒ t 1 t 2 = – 1
Now equation of circle will be (x – ct 1 ) (x – ct 2 ) + 
= 0
⇒ x 2 + y 2 – cx(t 1 + t 2 ) + yc(t 1 + t 2 ) – 2c 2 = 0
⇒ x 2 + y 2 – 2c 2 – c(t 1 + t 2 ) (x – y) = 0 ........(2)
equation (2) is of the form S + λ L = 0 where S is x 2 + y 2 – 2c 2 = 0 and L is x – y = 0
Solving S and L we get (c, c) and (–c, –c)
∴ (2) will always pass through (c, c) and (–c, –c)
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