How do you convert the following to the polar coordinates $ 2xy = 1 $ ?

Hint: In the given question, we are required to convert a Cartesian equation into polar equation by changing the Cartesian coordinates into polar coordinates. Cartesian coordinates can be easily changed to Polar coordinates by doing some minor tweaks, substitutions and changes. The polar coordinates system is a two dimensional coordinate system in which each point on a plane is determined by a distance from a reference point, generally origin and an angle from a reference position, generally x-axis.

Complete step by step solution:
The polar coordinates system deals with distance from origin and the angle made from positive x-axis.
The distance of a point from origin is denoted by r and is calculated as $ r = \sqrt {{x^2} + {y^2}} $ and the angle made by the vector joining the point with origin is denoted by $ \theta $ .
So, using trigonometry, we get,
 $ x = r\cos \theta $
 $ y = r\sin \theta $
So, replacing x by $ r\cos \theta $ and y by $ r\sin \theta $ in the Cartesian equation so as to convert t=it into an equation with polar coordinates.
So, we have,
 $ 2xy = 1 $
 $ \Rightarrow 2\left( {r\cos \theta } \right)\left( {r\sin \theta } \right) = 1 $
 $ \Rightarrow 2{r^2}\sin \theta \cos \theta = 1 $
 $ \Rightarrow {r^2}\left( {2\sin \theta \cos \theta } \right) = 1 $
 $ \Rightarrow {r^2}\left( {\sin 2\theta } \right) = 1 $
 $ \Rightarrow {r^2} = \dfrac{1}{{\sin \left( {2\theta } \right)}} $
 $ \Rightarrow {r^2} = \cos ec\left( {2\theta } \right) $
So, the equation $ 2xy = 1 $ when converted into polar coordinates becomes $ {r^2} = \cos ec\left( {2\theta } \right) $ .
So, the correct answer is “ $ {r^2} = \cos ec\left( {2\theta } \right) $ ”.

Note: We can directly convert the equation by making a couple of replacements like , replacing x by $ r\cos \theta $ and y by $ r\sin \theta $ in the Cartesian equation so as to convert t=it into equation with polar coordinates. These replacements directly convert the equation in cartesian coordinates into polar coordinates.