Question

How do you convert $xy = 1$ into polar form?

Answer 1

In this problem, we have given an equation in the rectangular coordinates. Here we are asked to convert the given rectangular form equation into the polar form. To convert the given equation into the polar form we need to use some identities and by using that identity we can get the required solution.

Formula used:
Half angle property of sine function: $\sin 2\theta = 2\sin \theta \cos \theta$

Complete step by step solution:
We are given the following equation in rectangular coordinates:
$xy = 1$…(i)
We are to find the equivalent polar coordinates.
The relation between rectangular coordinates $\left( {x,y} \right)$ and polar coordinates $\left( {r,\theta } \right)$ 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
$xy = 1$
$\Rightarrow \left( {r\cos \theta } \right)\left( {r\sin \theta } \right) = 1$
Multiply the terms, we get
$\Rightarrow {r^2}\cos \theta \sin \theta = 1$
Now, use the half angle property of the sine function, $\sin 2\theta = 2\sin \theta \cos \theta$ in the above equation.
$\Rightarrow {r^2} \times \dfrac{{\sin 2\theta }}{2} = 1$
Multiply both sides of the equation by $2$, we get
$\Rightarrow {r^2}\sin 2\theta = 2$

Hence, the required equivalent polar equation is ${r^2}\sin 2\theta = 2$.

Additional information:
In two dimensions, the Cartesian coordinate $\left( {x,y} \right)$ specifics the location of a point P in the plane. A polar coordinate system in a plane consists of a fixed point $O$, called the pole (or origin), and a ray emanating from the pole, called the polar axis. In such a coordinate system we can associate with each point $P$ in the plane a pair of polar coordinates $\left( {r,\theta } \right)$, where $r$ is the distance from $P$ to the pole and $\theta$ is an angle from the polar axis to the ray $OP$. The number $r$ is called the radical coordinate of $P$ and the number $\theta$ the angular coordinate (or polar angle) of $P$.

Note: In this problem our aim is to convert the given polar coordinate equation which is in the form $\left( {r,\theta } \right)$ into rectangular coordinates, for this we followed some important steps.
Relationship between Polar and Rectangular Coordinates:
Frequently, it will be useful to superimpose a rectangular $xy$-coordinate system on top of a polar coordinate system, making the positive $x$-axis coincide with the polar axis. If this is done, then every point P will have both rectangular coordinates $\left( {x,y} \right)$ and polar coordinates $\left( {r,\theta } \right)$. These coordinates are related by the equations
$x = r\cos \theta$, $y = r\sin \theta$…(1)
These equations are well suited for finding $x$ and $y$ when $r$ and $\theta$ are known. However, to find $r$ and $\theta$ when $x$ and $y$ are known, it is preferable to use the identities ${\sin ^2}\theta + {\cos ^2}\theta = 1$ and $\tan \theta = \sin \theta /\cos \theta$ to rewrite (1) as
${r^2} = {x^2} + {y^2}$, $\tan \theta = \dfrac{y}{x}$…(2