Question

(X-1)^2 + (y-3)^2=10 convert equation from rectangular form to polar form

Answer 1

`r= 2(cos(\theta) + 3sin(\theta))`

Explanation:

To convert an equation of rectangular shape to polar form use the following equivalences

`x=rcos(\theta)`

`y=rsin(\theta)`

`x^2 + y^2=r^2`

In this case we have the following equation.

`(x-1)^2 + (y-3)^2=10`

First we expanded the expression

`(x-1)^2 + (y-3)^2=10\\\\(x^2 -2x +1) +(y^2 -6y + 9) = 10\\\\\\x^2 +y^2 -2x -6y = 10-9-1\\\\x^2 +y^2 -2x -6y = 0`

Now we know that
`x^2 + y^2=r^2`,
`x=rcos(\theta)` and
`y=rsin(\theta)`

So

`r^2 -2rcos(\theta) -6rsin(\theta) = 0\\\\r^2 - 2r(cos(\theta) + 3sin(\theta))=0\\\\r^2 = 2r(cos(\theta) + 3sin(\theta))\\\\r= 2(cos(\theta) + 3sin(\theta))`