Question

Question #104ce

Answer 1

`y^2=3sqrt(x^2+y^2)+3x-x^2`

Explanation:

Use the following formulas (given by Pythagorean theorem, definition of sine, and definition of cosine, respectively):
`color(blue)(x^2+y^2=r^2)`
`color(blue)(y=rsintheta)`
`color(blue)(x=rcostheta)`

We have to manipulate the equation so that it is possible to substitute the formulas in.

(Note that the graph of `r=3+3costheta` is a cardioid, which is not a conic section, so the rectangular form of the equation will not be in any standard form.)

`r=3+3costheta`

`color(blue)(r^2)=3r+3color(blue)(rcostheta)`

`x^2+y^2=3sqrt(x^2+y^2)+3x`

`y^2=3sqrt(x^2+y^2)+3x-x^2`

Desmos