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If tangents O Q and O R from are drawn to variable circles having radius r and the centre lying on the rectangular hyperbola x y=1, then locus of circumcentre of \triangle O Q R is equal to... ( O is origin)
 

Option: 1

x y=4


Option: 2

x y=\frac{1}{4}


Option: 3

x y=1


Option: 4

None of these


Answers (1)

best_answer

Let S\left(t, \frac{1}{t}\right) be any point on the given rectangular hyperbola x y=1.


A circle is drawn with centre at S and radius r. From origin O, tangents OQ and OR are drawn to the above circle. OQSR is cyclic quadrilateral.
Hence, points O,Q,S and R are concyclic.
Circumcircle of \triangle OQR also passes through S and OS is the diameter.
Therefore, circumcentre of \triangle OQR is the mid-point of OS. If (x, y) is the circumcentre of \triangle OQR, then

\begin{array}{ll} & x=\frac{0+t}{2}, y=\frac{0+\frac{1}{t}}{2} \\ \therefore \quad & x y=\frac{1}{4} \end{array}

So, the required locus is x y=\frac{1}{4}.

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Sayak

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