Now, I've guessed the answer, it's a hyperbola, and I know what a hyperbolic function looks like, but I'm having a hard time getting it there. Here's my work so far:
- First off, $r^2 = x^2 + y^2$
- Therefore, $x^2 + y^2 = \frac{39}{\sin(2θ)}$
- Now, I'm thinking I should use the double angle formula: $\sin(2t) = 2\sin(t)\cos(t)$
- That way, I get $x^2 + y^2 = \frac{39}{2\sin(θ)\cos(θ)}$
- Now, $\sin(θ) = \frac{y}{r}$ and $\cos(θ) = \frac{x}{r}$
- This turns the above equation into $x^2 + y^2 = \dfrac{39}{2\cdot\frac{y}{r}\cdot\frac{x}{r}}$
- We can flip the r to the top... $x^2 + y^2 = \dfrac{39r}{x\cdot y}$
- $r = \sqrt{x^2 + y^2}$
- $x^2 + y^2 = \dfrac{39\sqrt{x^2 + y^2}}{xy}$
- ... I feel like I'm doing this wrong... because when I multiply this thing out I get: $2x^3y + 4x^3y^2 + 2xy^5 - 1521x^2 - 1521 y^2 = 0$... What am I doing wrong? I have a tendency to make things harder than they need to be.
Any help is greatly appreciated! I check this site kind of obsessively, so I'll respond quickly!