<s>[INST] Analytical solution to $\sqrt{a^2-x^2} + \sqrt{b^2-x^2} = \sqrt{a^2-x^2} \cdot \sqrt{b^2-x^2}$ Can't find beauty analytical solution to such equation :
$$\sqrt{a^2-x^2} + \sqrt{b^2-x^2} = \sqrt{a^2-x^2}\cdot\sqrt{b^2-x^2}$$
Assuming $a,\ b \in \mathbb{N},\ a \le b,\ x \in \mathbb{R}$
Is it possible to find a solution for general case?
 [/INST] Hint:
Let $\sqrt{a^2-x^2}=y$. We have
$$y+\sqrt{b^2-a^2+y^2}=y\sqrt{b^2-a^2+y^2},$$
$$y=(y-1)\sqrt{b^2-a^2+y^2}.$$
Squaring, you will obtain a quartic equation, which doesn't seem to simplify.
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