Given the rectangular equation xy=1, write an equivalent

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allisonlillian

01.01.2022

Mathematics

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

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

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