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:
${r^2} = {x^2} + {y^2}$
$x = r\cos \theta$
$y = r\sin \theta$