Circles Inside a Larger Circle

We note that the radius of the large circle is $10$ and that

$\begin{aligned} r + \sqrt{(r+5)^2-5^2} & = 10 \\ r + \sqrt{r^2 +10r} & = 10 \\ \sqrt{r^2 +10r} & = 10 - r \\ r^2 + 10 r & = 100-20r + r^2 \\ 30 r & = 100 \\ \implies r & = \frac {10}3 \approx \boxed 3 \end{aligned}$

We apply Descartes' theorem here. Let the radius of the pink circle be $r$ . Then by this theorem,

$\dfrac{1}{r}=-\dfrac{1}{10}+\dfrac{1}{5}+\dfrac{1}{5}-$

$2\sqrt {-\dfrac{1}{5\times 10}+\dfrac{1}{5\times 5}-\dfrac{1}{5\times 10}}=\dfrac{3}{10}$

$\implies r=\dfrac{10}{3}\approx 3.33$ .

So the required answer is $\boxed 3$ .

This was my latest YouTube Video Problem, I started this channel last week. I am open minded to any kind of comments for improvement purposes.

Here's the solution: Solution