Means of $a$ and $b$ given $(a-\sqrt3-\sqrt2)^2+(b-\sqrt3+\sqrt2)^2=0$

Question

Let $a,b\in\Bbb R$ so that $$(a-\sqrt3-\sqrt2)^2+(b-\sqrt3+\sqrt2)^2=0$$ Determine the arithmetic, geometric and harmonic means of $a$ and $b$.

When I open the parentheses, I obtain $a^2$ and $b^2$ so that the means cannot be determined.

Answer 1

Hint: Each summand is a square and hence $\ge 0$. For their sum to be equal to $0$ each summand (i.e. the term in each parenthesis) must be equal to zero.

Answer 2

$(x-a)^2+(y-b)^2=r^2$ determines a circle centred on $(a,b)$ with radius $r$. It follows that if $r=r^2=0$ the circle degenerates to a point, and thus the equation is only satisfied when $x=a$ and $y=b$. Hence $a=\sqrt3+\sqrt2$, $b=\sqrt3-\sqrt2$ and the means follow:

• Arithmetic mean is $\frac{\sqrt3+\sqrt2+\sqrt3-\sqrt2}2=\sqrt3$
• Geometric mean is $\sqrt{(\sqrt3)^2-(\sqrt2)^2}=1$
• Harmonic mean is $\frac2{\frac1{\sqrt3+\sqrt2}+\frac1{\sqrt3-\sqrt2}}=\frac1{\sqrt3}$