Prove that product of parameters of four concyclic points on the hyperbola `xy=c^(2)` is 1. Also, prove that the mean of these four concyclic points b

Monjur Alahi

Prove that product of parameters of four concyclic points on the hyperbola `xy=c^(2)` is 1. Also, prove that the mean of these four concyclic points bisects the distance between the centres of the hyperbola and the circle.


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Salim Ahmed Kawsar

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Given equation of hyperbola is `xy=c^(2)`
Let four conyclic point on the hyperbola be `(x_(i),y_(i)),i=1,2,3,4`.
Let the equation of the circle through point A, B, C and D be
`x^(2)+y^(2)+2gx+2fy+d=0" (1)"`
Solving circle and hyperbola, we get
`x^(2)+(c^(4))/(x^(2))+2gx+2f*(c^(2))/(x)+d=0`
`rArr" "x^(4)+2gx^(3)+dx^(2)+2fc^(2)x+c^(4)=0" (2)"`
`therefore" Product of roots,"x_(1)x_(2)x_(3)x_(4)=c^(4)`
Now, `(x_(i),y_(i))-=(ct_(i),(c)/(t_(i)))`
`therefore" "(ct_(1))(ct_(2))(ct_(3))(ct_(4))=c^(4)`
`rArr" "t_(1)t_(2)t_(3)t_(4)=1`
Now, mean of points `(x_(i),y_(i)),i=1,2,3,4," is "((sum_(i=1)^(4)x_(i))/(4),(sum_(i=1)^(4)y_(i))/(4))`.
From Eq. (2),
`x_(1)+x_(2)+x_(3)+x_(4)=-2g`
`rArr" "(sum_(i=1)^(4)x_(i))/(4)=-(g)/(2)`
Also, `sum_(i=1)^(4)y_(i)=sum_(i=1)^(4)(c^(2))/(x_(i))=(c^(2)sumx_(1)x_(2)x_(3))/(x_(1)x_(2)x_(3)x_(4))=(c^(2))/(c^(4))(-2fc^(2))=-2f`
`therefore" "(sumy_(i))/(4)=-(f)/(2)`
Thus, `((sum_(i=1)^(4)x_(i))/(4),(sum_(i=1)^(4)y_(i))/(4))-=(-(g)/(2),-(f)/(2))`.

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