Question

How do you convert $r = 2\cos \theta$ into rectangular form?

Answer 1

In the given question, we have been given the polar form of an equation. We have to convert it into its corresponding rectangular form. To do that, we convert the equation into the standard form where we have the variables in the standard form. Then we make the substitutions and find the answer.

Complete step by step answer:
We are going to use the following standard equations:
${r^2} = {x^2} + {y^2}$
$x = r\cos \theta$
We have to convert $r = 2\cos \theta$ into its rectangular form.
We are going to use the following standard equations:
${r^2} = {x^2} + {y^2}$ …(i)
$x = r\cos \theta$
Hence, $2x = 2r\cos \theta$ …(ii)
We have,
$r = 2\cos \theta$
Multiplying both sides by $r$,
${r^2} = 2r\cos \theta$
From (i) and (ii),
${x^2} + {y^2} = 2x$
Bringing all terms to one side,
$\left( {{x^2} - 2x} \right) + {y^2} = 0$
Adding $1$ to both sides,
$\left( {{x^2} - 2x + 1} \right) + {y^2} = 1$
We know, ${\left( {x - 1} \right)^2} = {x^2} - 2x + 1$

Hence, ${\left( {x - 1} \right)^2} + {y^2} = 1$ is the required equation.

Note: In the given question, we had to find the rectangular form of a polar form of an equation. We solved it by first converting the equation into its standard form. Then we made the substitutions by writing the formula and comparing them. The point where we could make a mistake is if we do not know the exactly correct formula, which would make us make wrong substitutions, and ultimately, make us write an incorrect equation.