Question

How do you write $r = 2a\cos \theta $ into a Cartesian equation?

Answer 1

Hint: We will first assume $x = r\cos \theta $ and $y = r\sin \theta $ and find r according to it. After that we will just put in the required values and modify to get the answer.

Complete step-by-step answer:
We have to write $r = 2a\cos \theta $ into a Cartesian equation.
Let us first of all assume that $x = r\cos \theta $ and $y = r\sin \theta $ in the above equation.
Since we assumed that $x = r\cos \theta $. On dividing this by r on both the sides, we will get:-
\[ \Rightarrow \dfrac{x}{r} = \dfrac{{r\cos \theta }}{r}\]
Is we cut off r from the right hand side in both numerator and denominator, we will then obtain the following expression:-
\[ \Rightarrow \dfrac{x}{r} = \cos \theta \] ……………(1)
Now, since we assumed that $x = r\cos \theta $ and $y = r\sin \theta $.
Squaring both, we will then obtain:-
$ \Rightarrow {x^2} = {\left( {r\cos \theta } \right)^2}$ and ${y^2} = {\left( {r\sin \theta } \right)^2}$
Simplifying the calculations by opening the squares in above expression, we will then obtain the following expression:-
$ \Rightarrow {x^2} = {r^2}{\cos ^2}\theta $ and ${y^2} = {r^2}{\sin ^2}\theta $
Adding both the expressions from the above line, we will get:-
\[ \Rightarrow {x^2} + {y^2} = {r^2}{\cos ^2}\theta + {r^2}{\sin ^2}\theta \]
Taking square of r common on the right hand side, we will then obtain:-
\[ \Rightarrow {x^2} + {y^2} = {r^2}\left( {{{\cos }^2}\theta + {{\sin }^2}\theta } \right)\]
Since, we know that \[{\cos ^2}\theta + {\sin ^2}\theta = 1\]. Using this in the above equation, we will then obtain the following expression:-
\[ \Rightarrow {x^2} + {y^2} = {r^2}\] …………………..(2)
Now, considering the equation given to us which is $r = 2a\cos \theta $.
Putting the equation (1) in it, we will then obtain:-
$ \Rightarrow r = 2a \times \dfrac{x}{r}$
Taking r from division in the right hand side to multiplication in the left hand side and re – arranging the terms to obtain the following expression:-
$ \Rightarrow x = 2a{r^2}$
Putting the equation (2) in the above expression to obtain the following expression:-
\[ \Rightarrow x = 2a\left( {{x^2} + {y^2}} \right)\]

Hence, we have the required Cartesian equation of $r = 2a\cos \theta $ as \[x = 2a\left( {{x^2} + {y^2}} \right)\].

Note:
The students must note that in the starting steps when we divided x by r in order to the required modification, we could divide because r is non zero because if r would have been zero, either a is zero which makes our equation as the point of origin or we have the angle theta equal to 90 degrees which is also very much fixed. Therefore, we have r is not equal to zero and we divided the whole equation by it. We can never divide any equation by 0 and we cannot multiply any equation by 0 as well.