Question

How do you convert $r = \dfrac{6}{{2 - 3\sin \theta }}$ into rectangular form?

Answer 1

Hint: Use the relation between polar and rectangular coordinates in order to convert the given polar form into rectangular form. First simplify the given equation by removing the fraction and then write it as $r$ at one side and left terms at opposite side, then use the relation $\sin \theta = \dfrac{y}{r}$ to remove sine function then square both sides, and as we know ${r^2} = {x^2} + {y^2}$ replace $r$ using this relation.

Formula used:
Relation between polar and rectangular coordinates: ${r^2} = {x^2} + {y^2}\;{\text{and}}\;\sin \theta = \dfrac{y}{r}$
Algebraic identity for square of sum of two numbers: ${(a + b)^2} = {a^2} + 2ab + {b^2}$

Complete step by step answer:
In order to convert the given polar form $r = \dfrac{6}{{2 - 3\sin \theta }}$ into rectangular form, we need to first simplify the given polar equation as follows
$\Rightarrow r = \dfrac{6}{{2 - 3\sin \theta }} \\
\Rightarrow r\left( {2 - 3\sin \theta } \right) = 6 \\
\Rightarrow 2r - 3r\sin \theta = 6 \\ $
Now, as we know from the relation between polar and rectangular coordinates that
$\sin \theta = \dfrac{y}{r} \Rightarrow r\sin \theta = y$
Using this relation to replace $r\sin \theta $ with $y$ we will get
$ \Rightarrow 2r - y = 6$
Further simplifying the equation and removing $y$ from left hand side by sending it to right hand side, we will get
$ \Rightarrow 2r = 6 + y$
Now squaring both sides, we will get
\[\Rightarrow {\left( {2r} \right)^2} = {\left( {6 + y} \right)^2} \\
\Rightarrow 4{r^2} = {\left( {6 + y} \right)^2} \\ \]
Expanding \[{\left( {6 + y} \right)^2}\] with the help of algebraic identity of square of sum of two numbers, which is given as ${(a + b)^2} = {a^2} + 2ab + {b^2}$, we will get
\[ \Rightarrow 4{r^2} = 36 + 12y + {y^2}\]
Now again from the relation between polar and rectangular coordinates we know that ${r^2} = {x^2} + {y^2}$, using this relation to convert the equation further, we will get
\[ \Rightarrow 4\left( {{x^2} + {y^2}} \right) = 36 + 12y + {y^2}\]
Simplifying it further,
\[\Rightarrow 4{x^2} + 4{y^2} = 36 + 12y + {y^2} \\
\therefore 4{x^2} + 3{y^2} - 12y - 36 = 0 \]

Note: If you only know the conversional relation for the coordinates between various planes (polar and rectangular plane), that’s enough to convert any equation of either form into another. As we have converted the equation of a curve from polar form to rectangular form, just use the relations smartly.