Question

How do you convert xy=16 into polar form?

Answer 1

Hint: This type of question is based on the concept of converting a linear equation with two variables to the polar form. Here, the rectangular coordinates are (x,y). We have to first convert the rectangular coordinates to polar coordinates \[\left( r,\theta \right)\] with the relation \[x=r\cos \theta \] and \[y=r\sin \theta \]. We then have to substitute the polar form of the coordinates x and y in the given equation. Multiply the whole equation by 2 and then use the trigonometric identity, that is, \[2\sin \theta \cos \theta =\sin 2\theta \] to simplify the polar form further.

Complete step-by-step solution:
According to the question, we are asked to xy=16 into polar form.
We have been given the equation is xy=16. ----------(1)
First, we have to consider the rectangular coordinates (x,y) which are related by the equation xy=16.
We know that the polar coordinates are \[\left( r,\theta \right)\].
The relation between rectangular coordinates (x,y) and polar coordinates is given by
\[x=r\cos \theta \] and \[y=r\sin \theta \].
Here, xy=16.
On equating the polar form of the rectangular coordinates in the given equation, we get
\[\left( r\sin \theta \right)\left( r\cos \theta \right)=16\]
On re-arranging the terms in the left-hand side of the equation, we get
\[\Rightarrow {{r}^{2}}\sin \theta \cos \theta =16\]
Let us now multiply 2 on both the sides of the equation.
\[\Rightarrow 2\times {{r}^{2}}\sin \theta \cos \theta =2\times 16\]
On further simplification, we get
\[\Rightarrow 2\times {{r}^{2}}\sin \theta \cos \theta =32\]
On rearranging the left-hand side of the equation, we get
\[\Rightarrow {{r}^{2}}\left( 2\sin \theta \cos \theta \right)=32\]
We know that \[2\sin \theta \cos \theta =\sin 2\theta \].
Therefore, we get
\[\Rightarrow {{r}^{2}}\left( \sin 2\theta \right)=32\]
\[\therefore {{r}^{2}}\sin 2\theta =32\]
Hence, the polar form of the equation xy=16 is \[{{r}^{2}}\sin 2\theta =32\].

Note: We should know the relation between polar coordinates and rectangular coordinates to solve this type of mistakes. We should not get confused by the substitution \[x=r\cos \theta \] and \[y=r\sin \theta \] not vice-versa. We should not make calculation mistakes based on sign conventions.