help understanding a weird phenomenon

Question

I was just thinking that you could closely calculate the square root of $10$ by averaging $10/3$ and $3$, and I put into desmos that equation with $2$ graphs: $y=\sqrt x$, $y=x/2a + a/2$. I saw that for all positive values of a there's some tangent point where the linear one is tangent to the curve. would love to know the explanation to that phenomenon, the relation between a and x or anything since I don't know how to figure it out. maybe its just the beginning of the Taylor series for $\sqrt{x}$?

Answer 1

Those two become tangent to each other at $x=a^{2}$ because their value are equal $\sqrt{a^{2}}=\frac{a^{2}}{2a}+\frac{a}{2}$ and their derivatives are also equal $\frac{1}{2\sqrt{a^{2}}}=\frac{1}{2a}$ for positive $a$

With this, You can make countless of such pair of equation e.g. $y=\sqrt{x}$ and $y=\frac{-a^{2}}{\sqrt{x}}+2a$ for positive real $a$