Question

How do you convert $xy = 4$ into polar form?

Answer 1

Hint: In converting the system to polar form we have to make use the formulas ${r^2} = {x^2} + {y^2}$, where $x = r\cos \theta $, and$y = r\sin \theta $, now substitute the values in the given equation, then make use of the trigonometric formula $2\sin \theta \cos \theta = \sin 2\theta $, we will get the required polar form.

Complete step by step solution:
To convert an equation given rectangular form (in $x$ and$y$) into in polar form (in the variables $r$and$\theta $) we will use the transformation relationships between the two sets of coordinates:
${r^2} = {x^2} + {y^2}$, where $x = r\cos \theta $, and$y = r\sin \theta $.
Polar coordinates are a complementary system to Cartesian coordinates, which are located by moving across an x-axis and up and down the y-axis in a rectangular fashion. While Cartesian coordinates are written as$\left( {x,y} \right)$, polar coordinates are written as$\left( {r,\theta } \right)$.
Now given equation is $xy = 4$,
Now using the relationship formulas,
$x = r\cos \theta $, and$y = r\sin \theta $,
By substituting these in then given equation we get,
$ \Rightarrow \left( {r\cos \theta } \right)\left( {r\sin \theta } \right) = 4$,
Now simplifying we get,
$ \Rightarrow {r^2}\sin \theta \cos \theta = 4$,
Now multiplying both sides with 2, we get,
$ \Rightarrow {r^2}2\sin \theta \cos \theta = 4 \times 2$,
Now using the trigonometric formula $2\sin \theta \cos \theta = \sin 2\theta $ we get,
$ \Rightarrow {r^2}\sin 2\theta = 8$,
So, the polar form is ${r^2}\sin 2\theta = 8$.


$\therefore $The polar form of the given rectangular form $xy = 4$ will be equal to ${r^2}\sin 2\theta = 8$.

Note:
In polar coordinates, a point in the plane is determined by its distance $r$ from the origin and the angle $\theta $ (in radians) between the line from the origin to the point and the x-axis. The polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.
In polar coordinates the origin is often called the pole. Because we aren't actually moving away from the origin/pole we know that. However, we can still rotate around the system by any angle we want and so the coordinates of the origin/pole are.