Question

How do you convert `r=1/2costheta` into cartesian form?

Answer 1

`(x - 1/4)^2 + y^2 = (1/4)^2`

Explanation:

the basic ideas are `x = r cos theta, y = r sin theta` so `x^2 + y^2 = r^2 cos^2 theta + r^2 sin^2 theta = r^2`

so using `r = sqrt{x^2 + y^2}` and `cos theta = x/r = x/sqrt{x^2 + y^2}` you can plug and play

So `sqrt{x^2 + y^2}= 1/2 x/sqrt{x^2 + y^2}`
`x^2 + y^2 = x/2`
`x^2 - x/2 + y^2 = 0`
completing the square in x
`(x - 1/4)^2 - 1/16 + y^2 = 0`
`(x - 1/4)^2 + y^2 = (1/4)^2`

that's a circle centred on (1/4, 0) radius 1/4.