Question

How do you convert `-x^2+2xy-y^2=9` into polar form?

Answer 1

`9/(sin2theta-1)`

Explanation:

using the formulae which link Cartesian to Polar coordinates

`• r^2 = x^2 + y^2`
`• x = rcostheta`
`• y = rsintheta`

rewrite as
`-x^2 - y^2 + 2xy = 9`

so -`(x^2 +y^2) + 2xy = 9`

hence `- r^2 + 2( rcostheta.rsintheta) =9`

and `-r^2 + 2r^2 costhetasintheta =9`

common factor: `r^2 (2costhetasintheta-1 )= 9`

(Note that : `sin2theta = 2costhetasintheta`)

hence `r^2(sin2theta - 1 ) = 9`

`rArr r^2 = 9/(sin2theta -1)`