Question

Write a polar equation of the parabola with focus at the origin and directrix y=2?

Answer 1

`r=-2/(1-sintheta)`

Explanation:

As the parabola is locus of a point which moves so that its distance from a point called focus and a line called directrix is always equal.

As focus is `(0,0)` and directrix is `y=2`, distance of a point `(x,y)` on the parabola from `(0,0)` and from `y=2` is equal i.e.

Further, the relation between Cartesial or rectangular coordinates `(x,y)` and polar coordinaates is given by `x=rcostheta` and `y=rsintheta`.

As equation of parabola is

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

we can write it as `r=rsintheta-2`

or `r=-2/(1-sintheta)`

Note that a negative `r` means at a distance of `r` in opposite direction i.e. at a distance of `r` positive at `pi+theta`.