MHB Polar equation of a conic (Carly's question at Yahoo Answers)

[Fernando Revilla]

Here is the question:

what is the conic represented by the polar equation r= 4 / (1 - 3 sin theta)
find the rectangular equation

Here is a link to the question:

R= 4 / (1 - 3 sin theta)? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[Fernando Revilla]

Hello Carly, follow the steps: $$\begin{aligned}&r=\frac{4}{1-3\sin \theta}\\& r(1-3\sin \theta)=4\\& r\left(1-3\dfrac{y}{r}\right)=4\\&r-3y=4\\& r=4+3y\\&r^2=(4+3y)^2\\&x^2+y^2=9y^2+24y+16\\&x^2-8y^2-24y-16=0\quad (*) \end{aligned}$$ The matrix of the conic is $A=\begin{bmatrix}{1}&{\;\;0}&{\;\;0}\\{0}&{-8}&{-12}\\{0}&{-12}&{-16}\end{bmatrix}$ and $\Delta=\det A\ne 0$, $\delta=\begin{vmatrix}{1}&{\;\;0}\\{0}&{-8}\end{vmatrix}<0$. This means that $(*)$ is the equation of a hyperbola. Alternatively, we can complete the squares.