Question

How do you convert $r = - 2\cos ec\theta $ into Cartesian form ?

Answer 1

Hint: In the given question, we are required to find an equation involving polar coordinates into an equation with Cartesian coordinates. Now, to convert polar coordinates into Cartesian coordinates, we have to replace x as $r\cos \theta $ and replace y as $r\sin \theta $ . Using these replacements, we can convert the polar coordinates into Cartesian coordinates with ease.

Complete step by step solution:
So, we have, $r = - 2\cos ec\theta $
Firstly, we take all the parameters to the left side of the equation and all the constants to the right side of the equation.
$ \Rightarrow \dfrac{r}{{\cos ec\theta }} = - 2$
Now, we know that $\cos ec\theta = \dfrac{1}{{\sin \theta }}$. So, we also have $\sin \theta = \dfrac{1}{{\cos ec\theta }}$. Hence, we get,
$ \Rightarrow r\sin \theta = - 2$
Now, we can replace $r\sin \theta $ with y so as to convert the equation given in polar coordinates into Cartesian coordinates. Now, simplifying further, we get,
$ \Rightarrow y = - 2$
So, the Cartesian form of the equation given to us in the polar coordinates as $r = - 2\cos ec\theta $ is $y = - 2$.

Note: For converting the polar coordinates into Cartesian coordinates, we replace the parameters of the polar coordinate system into Cartesian coordinate systems. So, we have x as $r\cos \theta $ and replace y as $r\sin \theta $. Also, when we square and add both the equations, we obtain the relation ${r^2}{\sin ^2}\theta + {r^2}{\cos ^2}\theta = {y^2} + {x^2}$ which can be further simplified as ${r^2} = {y^2} + {x^2}$ as we know that \[{\sin ^2}\theta + {\cos ^2}\theta = 1\] as a trigonometric identity. Therefore, the result ${r^2} = {y^2} + {x^2}$ can be used directly for solving such questions if required.