Question

How do you convert `x^2 - y^2 = 1` to polar form?

Answer 1

`r^2=sec 2theta`.

Explanation:

`x=r cos theta and y = r sin theta`. Also, use `cos^2theta-sin^2theta=cos 2theta`.

The given equations becomes `r^2 cos 2theta=1. So, r^2=sec 2theta`.

This represents a rectangular hyperbola with asymptotes along lines given by`r^2=oo=sec 2theta. So, theta = +- pi/4 and theta=+-(3pi)/4`, give the asymptotes as four half lines, from the the pole which is thw center of the hyperbola.

For a rectangular hyperbola, the semi- major axis a = semi-transverse axis b.
The eccentricity of a rectangular hyperbola = `sqrt 2`. .