Question

The polar equation of the form `x^2+2y-1=0` is .........?

Answer 1

`1/r=1+sintheta`

Explanation:

The relation between polar coordinates `(r,theta)` and rectangular coordinates `(x,y)` is given by `x=rcostheta` and `y=rsintheta`.

Hence `x^2+2y-1=0` can be written as

`r^2cos^2theta+2rsintheta-1=0`

or `r^2-r^2sin^2theta+2rsintheta-1=0`

or `r^2-(r^2sin^2theta-2rsintheta+1)=0`

or `r^2-(rsintheta-1)^2=0`

or `(r-rsintheta+1)(r+rsintheta-1)=0`

and either `r+rsintheta-1=0` i.e. `1/r=1+sintheta`

or `r-rsintheta+1=0` i.e. `1/r=sintheta-1`

In fact `1/r=1+sintheta` and `1/r=sintheta-1` both indicate same parabola (shown below).
graph{x^2+2y=1 [-5.167, 4.833, -3.36, 1.64]}