Question

How do you convert `y=3x-x^2` into polar form?

Answer 1

`x= r cos(theta), y = r sin(theta)`. Answer is `r = sec(theta) ( 3 - tan(theta))`.

Explanation:

The equation in rectangular form represents a parabola with vertex at (3/2, 9/4) and axis along negative y\axis. The parabola does not pass through the origin. So, r is never 0.

Answer 2

`r = sectheta(3- tantheta )`

Explanation:

using the formulae that links Cartesian to Polar coordinates.

`• x = rcostheta`

`• y = rsintheta`

the question can then be written as:

`rsintheta = 3rcostheta - r^2cos^2theta`

hence `r^2cos^2theta = 3rcostheta - rsintheta = r(3costheta - sintheta)`

(divide both sides by r )

hence `rcos^2theta =3 costheta - sintheta`

`rArr r =( 3costheta -sintheta)/cos^2theta = 3costheta/cos^2theta - sintheta/cos^2theta`

`= 3/costheta - tantheta/costheta = 3sectheta - tanthetasectheta`

`rArr sectheta (3 - tantheta )`