Question

How do you convert $ r = 4\cos \theta + 4\sin \theta $ into a cartesian equation $ ? $

Answer 1

Hint: The above equation is in the form of the polar equation, that is in terms of $ (r,\theta ) $ . When we have a polar equation and we require to convert it into a cartesian equation, that is in terms of $ (x,y) $ , we take $ x = r\cos \theta $ and $ y = r\sin \theta $

Complete step by step solution:
Given $ r = 4\cos \theta + 4\sin \theta - - - (1) $
Equation $ (1) $ which is in the polar form.
We have to convert the above polar equation in terms of the cartesian equation, that is the equation should be in terms of x and y only.
To do so, we shall make use of $ x = r\cos \theta $ and $ y = r\sin \theta $ .
Rearranging the above terms, we get,
 $
  \cos \theta = \dfrac{x}{r} \\
  \sin \theta = \dfrac{y}{r} \;
 $
Now, let us substitute equation $ (2) $ in equation $ (1) $ ,
 $ r = 4\left( {\dfrac{x}{r}} \right) + 4\left( {\dfrac{y}{r}} \right) $
 $ \Rightarrow r = \dfrac{{4(x + y)}}{r} $
Multiplying r on both sides we get,
 $ \Rightarrow r \times r = \dfrac{{r \times 4(x + y)}}{r} $
On the right hand side, r in numerator and denominator gets cancelled and we have the below equation,
 $ \Rightarrow {r^2} = 4(x + y) - - - (3) $
We know that in polar form, $ {r^2} = {x^2} + {y^2} $ ; put in the equation $ (3) $ we get,
 $ \Rightarrow {x^2} + {y^2} = 4(x + y) $
 $ \therefore {x^2} + {y^2} - 4(x + y) = 0 $
This is the required cartesian form of the given equation.
So, the correct answer is “ $ {x^2} + {y^2} - 4(x + y) = 0 $ ”.

Note:
$ \bullet $ To convert any equation, first we should identify in which form is the given equation.
 $ \bullet $ If the given equation is in terms of $ r's $ and $ \theta 's $ only, then it is in the polar form.
 $ \bullet $ If the given equation is in terms of x’s and y’s only, then it is in the cartesian form or rectangular form. Cartesian form is also known as rectangular form.
 $ \bullet $ If the given equation is in polar form, then look for $ {r^2},r\cos \theta ,r\sin \theta $
 $ \bullet $ If the given equation is in cartesian form, then look for $ {x^2} + {y^2},x,y $
 $ \bullet $ While converting the equation, we make use of these substitutions:
 $ {r^2} = {x^2} + {y^2} $
 $ x = r\cos \theta $
 $ y = r\sin \theta $