What is the first step I should take in solving this equation?

Question

I have to change this polar equation and put it in terms of $x$ and $y$.

$$r = \frac{5}{5\cos(\theta) + 6\sin(\theta)}$$

I was guessing that I should multiply all the terms by r and then convert from there, but is that the most efficient way to go? Just simple advice on how to start with this problem is sufficient for me. thank you in advance!

Answer 1

You want to get everything in terms of $x = r \cos(θ)$ and $y = r \sin(θ)$ so you do not need to multiply by $r$ since $1 = \frac{5}{ 5r \cos(θ) + 6 r \sin(θ) }$ and you are already quite done.

Answer 2

$$5r\cos\theta-5=30\sin\theta\cos\theta\ \ \ \ (1)$$

As $\dfrac x{\cos\theta}=\dfrac y{\sin\theta}=r,$

$(1)\implies$ $$5(x-1)=\dfrac{30xy}{r^2}\iff(x-1)=\dfrac{6xy}{x^2+y^2}$$

Answer 3

Since $x=r \cos \theta$ and $y=r \sin \theta$, therefore we can rewrite the given expression as $$5r \cos \theta + 6r \sin \theta=5.$$ Now we can use $x,y$ to get $$5x+6y=5.$$ (a straight line).