Simple geometry problem

Question

So I'm asking this question because I'm afraid I would be doing a stupid mistake... the problem associated with this trigonometry problem isn't pulling off. Could you tell me whether my calculation is correct?

In this figure:

$r,x_0,y_0$ are known. From that I have to calculate $\Delta r$.

My formula came out to be

$$\Delta r = y_0 + \sqrt{2r^{2}\left(1-\cos\left(\arcsin\left(\frac{x_{0}}{r}\right)\right)\right)-x_{0}^{2}}$$

Note: $y_0$ is the vertical dotted distance.

Answer 1

Let $a$ be the radius of the smaller circle. Then, by the pythagorean theorem: $$(a+y_0)^2+x_0^2 = r^2,$$ so: $$ a = -y_0+\sqrt{r^2-x_0^2}$$ and: $$\Delta r = r-a = r+y_0-\sqrt{r^2-x_0^2}.$$

Answer 2

$$\Delta r - y_0 +\sqrt{r^2-x_{0}^{2}}=r$$

Thus

$$\Delta r = r+ y_0 -\sqrt{r^2-x_{0}^{2}}$$