<s>[INST] 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
 [/INST] 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:
$$x^2+(y-2)^2=\left(\frac{9-2y}{3}\right)^2 \Rightarrow \\
9x^2+9y^2-36y+36=81-36y+4y^2 \Rightarrow \\
9x^2+5y^2=45 \Rightarrow \\
\frac{x^2}{5}+\frac{y^2}{9}=1.$$
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