Question

How do you write the given equation `x^2-y^2=1` into polar form?

Answer 1

`r^2=sec2theta`

Explanation:

Te relation between Cartesian coordinates `(x,y)` and polar coordinatess `(r,theta)` is given by

`x=rcostheta` and `y=rsintheta`

and hence `x^2-y^2=1` can be written as

`r^2cos^2theta-r^2sin^2theta=1`

or `r^2(cos^2theta-sin^2theta)=1`

or `r^2cos2theta=1`

or `r^2=sec2theta`

Note that when `theta` is between `45^@` and `135^@` as also between `225^@` and `315^@`, `sec2theta` is negative and hence in polar coordinates this curve does lie between these two regions.

graph{(x^2-y^2-1)(x+y)(x-y)=0 [-5, 5, -2.5, 2.5]}