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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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