Answer: The required equivalent polar equation is \(r^2\sin2\theta=2.\)
Step-by-step explanation: We are given the following equation in rectangular co-ordinates :
\(xy=1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)\)
We are to find the equivalent polar co-ordinates.
The relation between rectangular co-ordinates \((x,y)\) and polar co-ordinates \((r,\theta)\) is as follows :
\(x=r\cos \theta~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)\\\\y=r\sin\theta~~~~~~~~~~~~~~~~~~~~~~~~~~~(iii)\)
Substituting the values of x and y from equation (ii) and (iii) in equation (i), we get
\(xy=1\\\\\Rightarrow (r\cos\theta)(r\sin\theta)=1\\\\\Rightarrow r^2\cos\theta\sin\theta=1\\\\\Rightarrow \dfrac{r^2}{2}(2\cos\theta\sin\theta)=1\\\\\Rightarrow r^2\sin2\theta=2.\)
Thus, the required equivalent polar equation is \(r^2\sin2\theta=2.\)
