Question

Express r=4÷{3+4cos theta} as a Cartesian equation?

Answer 1

`7x^2-9y^2-32x+16=0`

Explanation:

The relation between polar coordinates `(r,theta)` and Cartesian coordinates `(x,y)` is given by `x=rcostheta`, `y=rsintheta` and `x^2+y^2=r^2`.

Hence `r=4/(3+4costheta)`

or `3r+4rcostheta=4`

or `3sqrt(x^2+y^2)+4x=4`

or `3sqrt(x^2+y^2)=4-4x`

or `9x^2+9y^2=16+16x^2-32x`

or `7x^2-9y^2-32x+16=0`

which is the equation of hyperbola.

graph{7x^2-9y^2-32x+16=0 [-10, 10, -5, 5]}