Please find the radius of the circle. Hello,
I want to find the radius of red circle. I tried it with several ways like trigonometric. But there is a special value is given with this. I can not understand why it is given. It is the blue colored value "2", the distance between two arcs of circles. Please help me to find this.
Thank you.
Strictly speaking, you do not need the information shown in blue,
because you can deduce it from the other information shown on the diagram.
Hence there is really no reason why that particular piece of 
information is given.

The important thing in this problem is that you can determine exactly
where the red circle intersects the $45$-degree diagonal line 
as shown in the diagram.
You know that the green circle has radius $20$ (because two of the radii
are explicitly labeled $20$), so you know the portion of the diagonal line
between the circumference of the green circle and the center of the green circle (the green dot) is $20$.
To determine exactly where the red circle intersects that line, you can
use the information that the distance from that point to the green dot
is $18$ (as shown on the diagram), or you can use the information
that the distance from that point to the circumference of the green
circle is $2$, which means the distance to the green dot is $18$.

The reason it is important to know where the red circle intersects the
diagonal line is that three points determine a circle.
You have two points on the red circle (where it intersects the green circle),
so this third point determines the center and radius of the red circle.

That is the abstract reason why the information in the diagram should be
sufficient to answer the question.
But as to exactly how to use that information, here are some hints:

Use the two perpendicular black lines as axes of a Cartesian coordinate system 
such that the green dot is at $(0,0)$ 
and the two intersections of the two circles are at $(0,20)$ and $(-20,0)$.
In that case, we can see that the red circle also passes through the point
$(-18\cos\frac{\pi}{4},18\sin\frac{\pi}{4}) = (-9\sqrt{2},9\sqrt{2})$;
that's where the red circle intersects the diagonal line in the upper left quadrant.

Let the coordinates of the red dot be $(x_0,y_0)$. Then $y_0 = -x_0$, 
so we can write the coordinates as $(x_0,-x_0)$. 
The distance from the red dot to $(0,20)$ is 
$\sqrt{x_0^2 + (-x_0 - 20)^2}$ 
and the distance from the red dot to $(-9\sqrt{2},9\sqrt{2})$ is
$\sqrt{(x_0 + 9\sqrt{2})^2 + (-x_0 - 9\sqrt{2})^2}$.
These two distances must be equal (because each is a radius of the red circle), that is,

$$\sqrt{x_0^2 + (-x_0 - 20)^2} = \sqrt{(x_0 + 9\sqrt{2})^2 + (-x_0 - 9\sqrt{2})^2}.$$

Solve for $x_0$. 
(Hint: simplify the last equation so that you can apply the quadratic formula.)
With that information, it is easy to find the radius of the circle.
Hint:

$R^{2}=20^{2}+(R-18)^{2}-2\cdot 20\cdot (R-18)cos(135)$

$R$=radius of the red circle.