Question

Given the rectangular equation xy=1, write an equivalent polar equation.

Answer 1

In your question where there is a rectangular equation xy=1, to make the polar equation of this equation is first you must substitute x=rcos(theta) and y=rsin(theta) after that the answer would be r2sinθcosθ=1r22sin(2θ)=1

Answer 2

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 $r^2\sin2\theta=2.$