Solving equations involving square roots I am a student and I often encounter these type of equations: 
$$\sqrt{x^2 + (y-2)^2} + \sqrt{x^2 + (y+2)^2} = 6$$
I usually solve these by taking one term ($\sqrt{x^2 + (y-2)^2}$ for example) to the right hand side but this seems to take more time. Please suggest me some methods which can help me solve these types of problem quickly.
Thanks

Denote:
$$\begin{cases}x^2 + (y-2)^2=t^2 \\ x^2+(y+2)^2=t^2+8y\end{cases}.$$
Then:
$$\sqrt{x^2 + (y-2)^2} + \sqrt{x^2 + (y+2)^2} = 6 \Rightarrow \\
t+\sqrt{t^2+8y}=6 \Rightarrow \\
t^2+8y=36-12t+t^2 \Rightarrow \\
t=\frac{9-2y}{3}.$$
Plug it into the first equation:
$$x^2+(y-2)^2=\left(\frac{9-2y}{3}\right)^2 \Rightarrow \\
9x^2+9y^2-36y+36=81-36y+4y^2 \Rightarrow \\
9x^2+5y^2=45 \Rightarrow \\
\frac{x^2}{5}+\frac{y^2}{9}=1.$$
