Question

How do you convert ${y^2} = 9x$ to polar form.

Answer 1

Hint: The conversion from rectangular to polar form is-
$x = r\cos \theta $
$y = r\sin \theta $
Use trigonometric identities such as $\dfrac{{\cos \theta }}{{\sin \theta }} = \cot \theta $ and $\dfrac{1}{{\sin \theta }} = \csc \theta $

Complete step by step solution:
As per the question convert ${y^2} = 9x$ to polar form is:
The conversion from rectangular to polar form is:
$x = r\cos \theta $
$y = r\sin \theta $
Given equation,
${y^2} = 9x$
Put the value $x = r\cos \theta $ and $y = r\sin \theta $ in the above equation to convert it into polar form.
Therefore,
${\left( {r\sin \theta } \right)^2} = 9\left( {r\cos \theta } \right)$
Here, ${\left( {r\sin \theta } \right)^2}$ will distribute the square after opening the bracket.
Therefore the modified equation will be,
${r^2}{\sin ^2}\theta = 9r\cos \theta $
In the above equation $9r\cos \theta $ is transferred to the right side and the sign will also change as it is shifted from right side to left side.
So,
${r^2}{\sin ^2}\theta - 9r\cos \theta = 0$
Take the common term, from the above equation.
$r\left( {r{{\sin }^2}\theta - 9\cos \theta } \right) = 0$
At this point either $r = 0$ or $r{\sin ^2}\theta - 9\cos \theta = 0$
Let’s solve the second one to get a meaningful answer.
As, $r = 0$
Therefore,
$r{\sin ^2}\theta - 9\cos \theta = 0$
Shift $' - 9\cos \theta '$ to right side
$\therefore r{\sin ^2}\theta = 9\cos \theta $
Write the value in terms of $'r'$
$r = \dfrac{{9\cos \theta }}{{{{\sin }^2}\theta }}$
Here, ${\sin ^2}\theta $ can be written as $\sin \theta \times \sin \theta $
Therefore,
$r = \dfrac{{9\cos \theta }}{{\sin \theta }}.\dfrac{1}{{\sin \theta }}$
As from the trigonometric identities.
$\dfrac{{\cos \theta }}{{\sin \theta }} = \cot \theta $ and $\dfrac{1}{{\sin \theta }} = \csc \theta $
After putting the above identities, the value of $'r'$ will be $r = 9\cot \theta .\csc \theta $

Hence, the polar form of ${y^2} = 9x$ is $r = 9\cot \theta .\csc \theta $

Additional information:
The polar form of a complex number is another way to represent a complex number. The form $z = a + bi$ is called the rectangular coordinate form of complex numbers.
The horizontal axis is the real axis and the vertical axis is the imaginary axis. You find the real and complex component in terms of $'r'$ and $'\theta '$ where $'r'$ is the length of the vector and $'\theta '$ is the angle made with the real axis.

Note: If you change the place of any number the sign will also get changed.
As,
$' + '$ Addition will convert into $' - '$ subtraction.
$' - '$ Subtraction will convert into $' + '$ Addition
$' \times '$ Multiplication will convert into $' \div '$ division
$' \div '$ division will convert into $' \times '$ multiplication.
Use trigonometric identities for converting into polar form.
For example: $\dfrac{{\cos \theta }}{{\sin \theta }} = \cot \theta ,\dfrac{1}{{\sin \theta }} = \csc \theta $